30PreCalc Scope and Sequence 3rd 4th - Blackboard Inc. and Sequence ... Math M:11-B:02 ... Which of the following expressions can be used to calculate the le ngth of an arc that subtends

PreCalculus 3rd and 4th Nine-Weeks

Scope and Sequence Topic 4: Trigonometry and Trigonometric Functions (45 – 50 days) A) Uses radian and degree angle measure to solve problems and perform conversions as needed. B) Uses the unit circle to explain the circular properties and periodic nature of trigonometric

functions and to find the trigonometric ratios of any angle. C) Describes and compares the characteristics of the trigonometric functions (with and without

the use of technology) for sine, cosine, tangent, cotangent, cosecant, and secant. D) Determines solutions to trigonometric equations. E) Describes how a change in the value of any constant in a general-form trigonometric

equation such as y = a sin (b-x) + c affects the graph of the equation. F) Represents the inverse of a trigonometric function symbolically and graphically G) Creates a scatterplot of bivariate data, identifies a trigonometric function to model the data,

and uses that model to identify patterns and make predictions. H) Derives and applies the basic trigonometric identities; i.e. angle addition, angle subtraction,

and double-angle. I) Uses trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines

and Law of Cosines. J) Models and solves problems using trigonometry. Topic 5: Noncartesian Representations (35 – 40 days) A) Uses vectors to model and solve application problems. B) Uses parametric equations to model and solve application problems. C) Uses polar coordinates. D) Expresses complex numbers in trigonometric form and computes sums, differences, products, quotients, powers, and roots of complex numbers in trigonometric form.

PreCalculus Standard 4 and 5 Trigonometry and Trigonometric Functions

Page 1 of 163 Columbus Public Schools 1/5/06

COLUMBUS PUBLIC SCHOOLS

MATHEMATICS CURRICULUM GUIDE

SUBJECT STATE STANDARD 3 and 4 TIME RANGE GRADING PreCalculus Patterns, Functions, and Algebra, 45-50 days PERIOD Geometry and Spatial Sense 3-4

MATHEMATICAL TOPIC 4 Trigonometry and Trigonometric Functions

CPS LEARNING GOALS A) Uses radian and degree angle measure to solve problems and perform conversions as needed. B) Uses the unit circle to explain the circular properties and periodic nature of trigonometric

functions and to find the trigonometric ratios of any angle. C) Describes and compares the characteristics of the trigonometric functions (with and without

the use of technology) for sine, cosine, tangent, cotangent, cosecant, and secant. D) Determines solutions to trigonometric equations. E) Describes how a change in the value of any constant in a general-form trigonometric

equation such as y = a sin (b-x) + c affects the graph of the equation. F) Represents the inverse of a trigonometric function symbolically and graphically G) Creates a scatterplot of bivariate data, identifies a trigonometric function to model the data,

and uses that model to identify patterns and make predictions. H) Derives and applies the basic trigonometric identities; i.e. angle addition, angle subtraction,

and double-angle. I) Uses trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines

and Law of Cosines. J) Models and solves problems using trigonometry.

COURSE LEVEL INDICATORS Course Level (i.e., How does a student demonstrate mastery?):

Converts between degrees and radians and can explain the appropriate use of each. Math M:11-B:02

Sets up and solves angular velocity and arc-length problems by using radian measure. Math G:12-D:02

Describes and compares the characteristics of the trigonometric functions; e.g., general shape, number of roots, domain and range, even or odd asymptotic and global behavior for sine, cosine, tangent, cotangent, cosecant, and secant both algebraically and graphically. Math A:12-A:03

Relates a given sine or cosine graph to its equivalent other in terms of phase shift. Math A:12-A:03

Determines all zeros for a trigonometric function algebraically and also gives all zeros within a given range (such as between 0 and 2π radians). Math A:12-A:03

Identifies the amplitude, frequency/period, phase shift, vertical shift, etc. of a given trigonometric function and uses these to sketch a graph of the function. Math A:12-A:03

Identifies the extrema of trigonometric functions with and without technology. Math A:12-A:03



Uses the unit circle to explain the circular properties and periodic nature of trigonometric functions and to find the trigonometric ratios of any angle. Math G:11-A:04

Uses the unit circle to explain the periodic nature of the sine and cosine functions, the nature of reference angles, and the range of possible values for sine and cosine. Math A:12-A:03

Represents the inverse of a trigonometric function symbolically and graphically. Math A:12-A:04

Plots bivariate data and determines the trigonometric function that best fits the data both analytically and by regression. Math D:11-A:04

Determines phase shift, vertical shift, amplitude and frequency to be able to create the trigonometric function equation best fitting the data. Math A:12-A:03

Collects real world motion data and models it using trigonometric equations. Math D:11-C:04

Verifies identities analytically by applying fundamental trigonometric identities to re-write and combine expressions. Math G:12-A:02

Verifies trigonometric identities graphically. Math G:12-A:02 Uses the double-angle, half-angle, and angle-addition formulas to determine specified

trigonometric values. Math G:11-A:04 Uses trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines

and Law of Cosines. Math G:11-A:04 Uses the Law of Sines and the Law of Cosines to determine the missing angles or sides of

triangles. Math G:11-A:04 Uses Heron’s formula to find the area of a triangle when the sides are known but base and/or

height are not given. Math G:11-A:04 Determines general solutions to trigonometric equations and specific solutions within a given

interval. Math G:11-A:04 Previous Level:

Defines the basic trigonometric ratios in right triangles: sine, cosine, and tangent. Math G:09-I:01

Uses right triangle trigonometric relationships to determine lengths and angle measures. Math G:09-I:02

Evaluates expressions containing square roots. Math N:09-I:04 Sketches a basic sine and cosine graph by hand. Math A:10-D:02 Sketches the graph of a function by means of technology. Math A:08-D:09

Next Level:

Determines the average rate of change for specific trigonometric functions. Math A:12-A:10



*This is an extension of the benchmarks in grades 8-10 for more complex figures.

The description from the state, for the Geometry and Spatial Sense Standard says: Students identify, classify, compare and analyze characteristics, properties and relationships of one-, two-, and three-dimensional geometric figures and objects. Students use spatial reasoning, properties of geometric objects and transformations to analyze mathematical situations and solve problems. The grade-band benchmarks from the state, for this topic in the grade band 11 – 12 are: A. Use trigonometric relationships to verify and determine solutions in problem situations. D. Use coordinate geometry to represent and examine the properties of geometric figures.*

The description from the state, for the Patterns, Functions, and Algebra Standard says: Students use patterns, relations, and functions to model, represent and analyze problem situations that involve variable quantities. Students analyze, model and solve problems using various representations such as tables, graphs and equations. The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is: A. Analyze functions by investigating rates of change, intercepts, zeros, asymptotes and local

and global behavior.

The description from the state, for the Mathematical Processes Standard says: Students use mathematical processes and knowledge to solve problems. Students apply problem-solving and decision-making techniques, and communicate mathematical ideas. The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is: J. Apply mathematical modeling to workplace and consumer situations including problem

formulation, identification of a mathematical model, interpretation of solution within the model, and validation to original problem situation.

The description from the state for the Measurement Standard says: Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools and technologies. The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is: B. Apply various measurement scales to describe phenomena and solve problems.

The description from the state for the Data Analysis Standard says: Students pose questions and collect, organize, represent, interpret and analyze data to answer those questions. Students develop and evaluate inferences, predictions and arguments that are based on data. The grade-band benchmarks from the state, for this topic in the grade band 11 – 12 are: A. Create and analyze tabular and graphical displays of data using appropriate tools, including

spreadsheets and graphing calculators. C. Design and perform a statistical experiment, simulation or study; collect and interpret data;

and use descriptive statistics to communicate and support predictions and conclusions.



PRACTICE ASSESSMENT ITEMS

A wheel rotating at 50 revolutions per minute rotates at A. 50π radians/minute B. 100π radians/minute C. 150π radians/minute D. 200π radians/minute

Which of the following expressions can be used to calculate the length of an arc that subtends a central angle of measure 65º on a circle of diameter 30 meters?

A. 65 30180º

π× × meters

B. 18065 30π

× × meters

C. 65 30× meters D. 65 meters

Trigonometry - A

Which of the following gives the measures of two angles, one positive and one negative, that are coterminal with a 73 degree angle? A. 433o, -287o B. 433o, -107o C. 163o, -17o D. 253o, -107o




Answers/Rubrics

Low Complexity Moderate Complexity

Trigonometry - A

A wheel rotating at 50 revolutions per minute rotates at A. 50π radians/minute B. 100π radians/minute C. 150π radians/minute D. 200π radians/minute

Answer: B

Which of the following gives the measures of two angles, one positive and one negative, that are coterminal with a 73 degree angle? A. 433o, -287o B. 433o, -107o C. 163o, -17o D. 253o, -107o

Answer: A

Which of the following expressions can be used to calculate the length of an arc that subtends a central angle of measure 65º on a circle of diameter 30 meters?

A. 65 30180º

π× × meters

B. 18065 30π

× × meters

C. 65 30× meters D. 65 meters

Answer: A




Trigonometry - A

A neighborhood carnival has a Ferris wheel whose radius is 30 feet. You measure that it takes

70 seconds to complete one revolution. Which of the following correctly determines the angular

speed ω and the linear speed v of the Ferris wheel?

A. ω =

θt

=πrad70sec

= .045rad / sec , v =

st

=2π × 30 ft

70sec= 2.7 ft / sec

B. ω =

θt

=2πrad70sec

= .09rad / sec , v =

st

=30 ft

70sec= 0.43 ft / sec

C. ω =

θt

=30 ft

70sec= 0.43 ft / sec ,

v =

st

=30 ft


D. ω =

θt

=2πrad70sec

= .09rad / sec , v =

st

=2π × 30 ft

70sec= 2.7 ft / sec

A water sprinkler sprays water over a distance of 30 feet while rotating through an angle of 135º a. Determine the area of the lawn that receives water.

b. If we wanted the area of coverage to be exactly 1500 square feet, determine we should change the radius to, keeping the angle the same.

It takes ten identical pieces to form a circular track for a pair of toy racing cars. If the inside arc of each piece is 5.8 inches shorter than the outside arc, determine the width of the track.




Answers/Rubrics

High Complexity Short Answer/Extended Response

Trigonometry - A

A neighborhood carnival has a Ferris wheel whose radius is 30 feet. You measure that it takes

70 seconds to complete one revolution. Which of the following correctly determines the angular

speed ω and the linear speed v of the Ferris wheel?

A. ω =

θt

=πrad70sec

= .045rad / sec , v =st

=2π × 30 ft

70sec= 2.7 ft / sec

B. ω =

θt

=2πrad70sec

= .09rad / sec , v =

st

=30 ft


C. ω =

θt

=30 ft

70sec= 0.43 ft / sec ,

v =

st

=30 ft


D. ω =

θt

=2πrad70sec

= .09rad / sec , v =

st

=2π × 30 ft

70sec= 2.7 ft / sec

Answer: D

A water sprinkler sprays water over a distance of 30 feet while rotating through an angle of 135º a. Determine the area of the lawn that receives water. b. If we wanted the area of coverage to be exactly 1500 square feet, determine we should change the radius to, keeping the angle the same. Solution

a. 3135180 4radians radiansπ π× =

2 2 231 12 2 430 1060.29A r ftπθ= = ⋅ =

b. 2 2 231 12 2 4 1500 1273.24 35.68A r r r r ftπθ= = = =

A 4 point response: shows work and gets correct answers to both parts. A 3 point response gets one solution correct, and the other has a single error in

substitution or calculation. A 2 point response gets the first part right, but does not get the proper set-up for the

second part. A 1 point response has a major conceptual error on the first part, and does not show

any understanding of the second part. A 0 point response shows no understanding of the problem.




Answers/Rubrics

Short Answer/Extended Response

Trigonometry - A

It takes ten identical pieces to form a circular track for a pair of toy racing cars. If the inside arc of each piece is 5.8 inches shorter than the outside arc, determine the width of the track. Solution:

x+5.8 x

w

r

Each piece of track has an angle measure of 210π⎛ ⎞

⎜ ⎟⎝ ⎠

(or 5π⎛ ⎞

⎜ ⎟⎝ ⎠

) radians.

The arc length of the inside of one piece we’re taking as x, so x = 5π⎛ ⎞

⎜ ⎟⎝ ⎠

r

The arc length of the outside of one piece will then be: x + 5.8 = 5π⎛ ⎞

⎜ ⎟⎝ ⎠

(r + w)

To solve both of these equations simultaneously, we multiply both equations by 5 and get: 5x = πr and 5x + 29 = π(r + w)

Solving these, we get w = 29π

⎛ ⎞⎜ ⎟⎝ ⎠

= 9.23 inches

A 4-point response uses the arc-length formulas to set-up and solve simultaneous equations and gets the correct width. A 3-point response has appropriate set-up for the two arc-length formulas but doesn’t reach a proper solution for the width. A 2-point response specifies the arc-length formula but doesn’t properly represent the 5.8-inch difference and does not determine the width. A 1-point response illustrates all necessary components of the problem but goes no further. A 0-point response shows no understanding of the problem.




Which of the following functions have identical graphs? (i)

y = sin 2x + π

6( )

(ii) y = cos 2x − π6( )

(iii) y = cos 2x − π

3( ) A) i and iii B) ii and iii C) i and ii D) i, ii and iii

Trigonometry - B

Which of the following gives the range of the function f(x) = (sin x)2+(cos x)2 ? A. {0} B. {1} C. [0, 1] D. [–1, 1] E. [0, 2] A sinusoid with amplitude 6 has a maximum value of 3. What is its minimum value? A. -12 B -10 C. -9 D. 0 E. 3




Answers/Rubrics


Trigonometry - B

Which of the following gives the range of the function f(x) = (sin x)2+(cos x)2 ? A. {0} B. {1} C. [0, 1] D. [–1, 1] E. [0, 2]

Answer: B A sinusoid with amplitude 6 has a maximum value of 3. What is its minimum value? A. -12 B. -10 C. -9 D. 0 E. 3

Answer: C

Which of the following functions have identical graphs? (i)

y = sin 2x + π

6( )

(ii) y = cos 2x − π6( )

(iii) y = cos 2x − π

3( ) A) i and iii B) ii and iii C) i and ii D) i, ii and iii

Answer: A




Trigonometry - B

Which of the following trigonometric functions are odd? i. y = sin(x) ii. y = cos(x) iii. y = tan(x) A. i and ii only B. ii and iii only C. i and iii only D. i, ii, and iii

Based on years of weather data in a certain city, the expected low temperature T (in ºF) can be

approximated by T = 40sin

2π365

t −3814

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ + 15 where t is in days with t = 0 corresponding to

January 1. Predict the date when the coldest day of the year will occur and give the temperature for that day. Show your solution.




Answers/Rubrics


Trigonometry - B

Which of the following trigonometric functions are odd? i. y = sin(x) ii. y = cos(x) iii. y = tan(x) A. i and ii only B. ii and iii only C. i and iii only D. i, ii, and iii

Answer: C

Based on years of weather data in a certain city, the expected low temperature T (in ºF) can be

approximated by T = 40sin

2π365

t −3814

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ + 15 where t is in days with t = 0 corresponding to

January 1. Predict the date when the coldest day of the year will occur and give the temperature for that day. Show your solution. Solution: By using the minimum function on the graphing calculator, we get minimum temperatures of -25o on day 4 (January 4). A 2-point response discusses proper use of the graph to get the correct answer. A 1-point response uses a graph to get an incorrect answer. A 0-point response shows no mathematical understanding of the task.




π6

is the reference angle for which one of the following non-acute angles?

A)

π3

B)

2π3

C)

5π6

D)

4π3

Which of the following must have the same value as cos(68º)? A. sin(22º) B. cos(22º) C. tan(22º) D. none of the above

Trigonometry - C




Answers/Rubrics


Trigonometry - C

π6

is the reference angle for which one of the following non-acute angles?

A)

π3

B)

2π3

C)

5π6

D)

4π3

Answer: C

Which of the following must have the same value as cos(68º)? A. sin(22º) B. cos(22º) C. tan(22º) D. none of the above

Answer: A




Trigonometry - C

If cosθ =

1213

and tanθ < 0 , then sinθ = ___?

A. −

1213

B.

−512

C.

513

D. −

513

What would be the coordinates of the point P where the terminal ray of an angle θ intersects the unit circle? A. (sinθ ,cosθ) B. (cosθ ,sinθ) C. (r, r2) D. (r 2 ,θ)

r=1

P

θ




Answers/Rubrics

High Complexity High Complexity

Trigonometry - C

If cosθ =

1213

and tanθ < 0 , then sinθ = ___?

A. −

1213

B.

−512

C.

513

D. −

513

Answer: D

What would be the coordinates of the point P where the terminal ray of an angle θ intersects the unit circle? A. (sinθ ,cosθ) B. (cosθ ,sinθ) C. (r, r2) D. (r 2 ,θ)

Answer: B

r=1

P

θ




Evalute cos

7π6

without using a calculator by using ratios and the relevant reference triangle.

Show all work.

Trigonometry - C




Answers/Rubrics


Trigonometry - C

Evalute cos

7π6

without using a calculator by using ratios and the relevant reference triangle.

Show all work. Solution: 7π6

is in the third quadrant where cosine is negative. The reference angle for the third

quadrant is 7π6

− π , or π6

. Thus we need − cosπ6

which equals −3

2

A 2-point response properly explains where we get the sign of the answer and how we determine the reference angle, and gets the correct answer. A 1-point response makes at most one error (sign, reference angle, or result of trig function). A 0-point response shows no mathematical understanding of the task.




Which one of the following trigonometric functions has its graph symmetric about the line y = - 6? A) y = −6sin(3x) B) y = 8sin(x − 6) + 1 C) y = 4 cos(2x) − 6

D) y = 2sinx

−6⎛⎝⎜

⎞⎠⎟

What is the phase shift of y = 5sin(2x − 3π ) ? A) 3π to the left B) 3π to the right

C) 3π2

to the left

D) 3π2

to the right

Trigonometry - D




Answers/Rubrics


Trigonometry - D

Which one of the following trigonometric functions has its graph symmetric about the line y = - 6? A) y = −6sin(3x) B) y = 8sin(x − 6) + 1 C) y = 4 cos(2x) − 6

D) y = 2sinx

−6⎛⎝⎜

⎞⎠⎟

Answer: C

What is the phase shift of y = 5sin(2x − 3π ) ? A) 3π to the left B) 3π to the right

C) 3π2

to the left

D) 3π2

to the right

Answer: C (Trick: This is really y = 5sin2(x - 32π )



-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-4

-3

-2

-1

1

2

3

4


Trigonometry - D

Which equation matches the graph below? A)

y = −2cos4 x − π( )

B)

y = −2cos 4x − π( )

C)

y = −2cos4 x + π( )

D)

y = −2cos 4x + π( )

State the amplitude and period of the sinusoid and (relative to the parent function) the phase shift

and vertical shift: y = 6cos

x + 42

⎛⎝⎜

⎞⎠⎟

− 3 .



-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-4

-3

-2

-1

1

2

3

4


Answers/Rubrics


Trigonometry - D

Which equation matches the graph below? A)

y = −2cos4 x − π( )

B)

y = −2cos 4x − π( )

C)

y = −2cos4 x + π( )

D)

y = −2cos 4x + π( )

Answer: D

State the amplitude and period of the sinusoid and (relative to the parent function) the phase shift

and vertical shift: y = 6cos

x + 42

⎛⎝⎜

⎞⎠⎟

− 3 .

Solution: Cosine graph, shifted left 4, multiply period by 2 (horizontal stretch, yielding a period of 2π

12

or 4π , multiply amplitude by 6, and shift down by 3.

A 2 point response correctly determines all information asked for. A 1 point response gets at least 2 pieces of the question correct but shows a lack of understanding for the other two. A 0 point response shows no mathematical understanding of the problem.




The exact value of arctan 3 is: A. 0

B.

π6

C.

π4

D.

π3

The range of the function f (x) = arcsin x is: A. (−∞,∞) B. (−1,1) C. [−1,1] D. [0,π ] E. [ ]/ 2, / 2π π−

Trigonometry - E




Answers/Rubrics


Trigonometry - E

The exact value of arctan 3 is: A. 0

B.

π6

C.

π4

D.

π3

Answer: D

The range of the function f (x) = arcsin x is: A. (−∞,∞) B. (−1,1) C. [−1,1] D. [0,π ] E. [ ]/ 2, / 2π π−

Answer: E




Trigonometry - E

sec(tan−1(x)) = A. x B. csc x

C. 1+ x2

D. 1− x2

E.

sin xcos2 x

You rent a cottage on the ocean for a week one summer and notice that the tide comes in twice daily with approximate regularity. Remembering that the trigonometric functions model repetitive behavior, you place a meter stick in the water to measure water height every hour between 6:00 AM to midnight. At low tide the height of the water is 0 centimeters and at high tide the height is 80 centimeters. Determine a reasonable defining equation for this function and explain how you determined your answer.




Answers/Rubrics


Trigonometry - E

sec(tan−1(x)) = A. x B. csc x

C. 1+ x2

D. 1− x2

E.

sin xcos2 x

Answer: C

You rent a cottage on the ocean for a week one summer and notice that the tide comes in twice daily with approximate regularity. Remembering that the trigonometric functions model repetitive behavior, you place a meter stick in the water to measure water height every hour between 6:00 AM to midnight. At low tide the height of the water is 0 centimeters and at high tide the height is 80 centimeters. Determine a reasonable defining equation for this function and explain how you determined your answer. Student Answers Will Vary. A sample solution is:

Amplitude: a = 1 (80 0)2

− = 40. Period: 2 12bπ = , therefore b =

6π .

Displacement: 3 = cb− , therefore c =

2π− . Vertical Shift = d = 40.

y = 40 sin6 2

xπ π⎛ ⎞−⎜ ⎟⎝ ⎠

+ 40

A 2 point response correctly determines all information asked for. A 1 point response gets at least 2 pieces of the question correct but shows a lack of understanding for the other two. A 0 point response shows no mathematical understanding of the problem.



-6.28319 -3.14159 3.14159 6.28319

-4

-3

-2

-1

1

2

3

4

-6.28319 -3.14159 3.14159 6.28319

-4

-3

-2

-1

1

2

3

4

-6.28319 -3.14159 3.14159 6.28319

-4

-3

-2

-1

1

2

3

4

-6.28319 -3.14159 3.14159 6.28319

-4

-3

-2

-1

1

2

3

4


Which statement is always true for the inverse of every trigonometric function? A. The inverse of every trigonometric function has a range of [-1, 1]. B. The inverse of every trigonometric function passes the vertical line test. C. The inverse of every trigonometric function has a domain of [0, 2π]. D. The inverse of every trigonometric function is not a function.

Which graph represent the equation y = cos-1x? A. B. C. D.

Trigonometry - F



-6.28319 -3.14159 3.14159 6.28319

-4

-3

-2

-1

1

2

3

4

-6.28319 -3.14159 3.14159 6.28319

-4

-3

-2

-1

1

2

3

4

-6.28319 -3.14159 3.14159 6.28319

-4

-3

-2

-1

1

2

3

4

-6.28319 -3.14159 3.14159 6.28319

-4

-3

-2

-1

1

2

3

4


Answers/Rubrics


Trigonometry - F

Which statement is always true for the inverse of every trigonometric function? A. The inverse of every trigonometric function has a range of [-1, 1]. B. The inverse of every trigonometric function passes the vertical line test. C. The inverse of every trigonometric function has a domain of [0, 2π]. D. The inverse of every trigonometric function is not a function.

Answer: D

Which graph represent the equation y = cos-1x? A. B. C. D.

Answer: C




Trigonometry - F

Suppose that the average monthly low temperatures (rounded to the nearest degree) for a small town are shown in the table. Month 1 2 3 4 5 6 7 8 9 10 11 12 13 Temp (oF)

59 63.4 65 63.4 59 53 47 42.6 41 42.6 47 53 59

Model this data using f(x) = a cos(b(x-c)) + d A.

f (x) = 16cos π

6 (x − 3)( )+ 53 B.

f (x) = 12cos π

6 (x − 3)( )+ 53 C.

f (x) = 16cos π

6 x( )+ 53 D.

f (x) = 16cos π

6 (x + 3)( )+ 53

Tides go up and down in a 12.2-hour period. The average depth of a certain river is 14m and ranges from 9 to 19m. The variation can be approximated by a sine curve. Write an equation that gives the approximate variation y, if x is the number of hours after midnight with high tide occurring at 8 am. Justify each part of your answer using relevant terminology.




Answers/Rubrics

High Complexity

Trigonometry - F

Suppose that the average monthly low temperatures (rounded to the nearest degree) for a small town are shown in the table. Month 1 2 3 4 5 6 7 8 9 10 11 12 13 Temp (oF)

59 63.4 65 63.4 59 53 47 42.6 41 42.6 47 53 59

Model this data using f(x) = a cos(b(x-c)) + d A.

f (x) = 16cos π

6 (x − 3)( )+ 53 B.

f (x) = 12cos π

6 (x − 3)( )+ 53 C.

f (x) = 16cos π

6 x( )+ 53 D.

f (x) = 16cos π

6 (x + 3)( )+ 53

Answer: A




Answers/Rubrics


Trigonometry - F

Tides go up and down in a 12.2-hour period. The average depth of a certain river is 14m and ranges from 9 to 19m. The variation can be approximated by a sine curve. Write an equation that gives the approximate variation y, if x is the number of hours after midnight with high tide occurring at 8 am. Justify each part of your answer using relevant terminology. Answer:

The period is 12.2, so the coefficient for x would be 212.2

π , or 6.1π .

The average depth of 14 gives our vertical shift of 14. The starting point being hour 8 means that 8 must be subtracted from x in the equation (phase shift of 8 hours to the left from the parent graph) Half the difference between our maximum, 19m, and our minimum, 9 m, gives us our amplitude of 5.

( )5sin 8 146.1

y xπ⎛ ⎞= − +⎜ ⎟⎝ ⎠

A 4 point response includes proper explanations for the period, vertical shift, period, phase shift, and amplitude, and puts them together into the equation given above. A 3 point response has one mistake in one of the four properties described above but shows a proper understanding of the other three. A 2 point response has flaws in two of the four modifications to the parent function, but properly represents the other two in the equation. A 1 point response shows understanding of one aspect of the data in terms of modifying the parent function. A 0 point response shows no mathematical understanding of the problem.




If 4sin( )5

x = and 3cos( )5

x = , then sin(2x) and cos(2x) would be:

A. 24 7sin(2 ) , cos(2 )25 25

x x= = −

B. 12 1sin(2 ) , cos(2 )25 5

x x= = −

C. 8 6sin(2 ) , cos(2 )5 5

x x= =

D. 16 9sin(2 ) , cos(2 )25 25

x x= =

Which one of the following is equivalent to tan secθ θ+ ? A. sin cosθ θ+ B. tan cscθ θ+

C. sin 1cos

θθ+

D. cos1 sin

θθ+

E. cos cotθ θ−

Trigonometry - G




Answers/Rubrics


Trigonometry - G

If 4sin( )5

x = and 3cos( )5

x = , then sin(2x) and cos(2x) would be:

A. 24 7sin(2 ) , cos(2 )25 25

x x= =

B. 12 1sin(2 ) , cos(2 )25 5

x x= = −

C. 8 6sin(2 ) , cos(2 )5 5

x x= =

D. 16 9sin(2 ) , cos(2 )25 25

x x= =

Answer: A

Which one of the following is equivalent to tan secθ θ+ ? A. sin cosθ θ+ B. tan cscθ θ+

C. sin 1cos

θθ+

D. cos1 sin

θθ+

E. cos cotθ θ−

Answer: C




Trigonometry - G

Which of the following gives the expression cos cos1 cos 1 cos

x xx x

−+ −

in completely simplified form?

A. 2 csc2x B. -2cot2x C. -2tan2x D. 2sec2x

Prove the identity: 2(cos )(tan sin cot ) sin cosx x x x x x+ = +




Answers/Rubrics


Trigonometry - G

Which of the following gives the expression cos cos1 cos 1 cos

x xx x

−+ −

in completely simplified form?

A. 2 csc2x B. -2cot2x C. -2tan2x D. 2sec2x

Answer: B

Prove the identity: 2(cos )(tan sin cot ) sin cosx x x x x x+ = + Answers Will Vary. A sample solution is:

2

2

2 2

sin cos(cos ) sin sin coscos sinsin(cos ) cos sin coscos

sin cos sin cos

x xx x x xx xxx x x xx

x x x x

⎛ ⎞+ • = +⎜ ⎟⎝ ⎠⎛ ⎞+ = +⎜ ⎟⎝ ⎠

+ = +




Two boats starting at the same place and time, speed away along courses that form a 150º angle. If one boat travels at 54 miles per hour and the other boat travels at 30 miles per hour, determine how far apart the boats are after 20 minutes? A. 15.62 mi B. 18.43 mi C. 24.08 mi D. 27.13 mi E. 579.88 mi

Given a triangle with the following information provided: a = 38, b = 19, C = 122º, which of the following would we use first in order to solve the triangle: A. The Pythagorean Theorem B. The Law of Sines C. The Law of Cosines D. The Quadratic Formula

Trigonometry - H




Answers/Rubrics


Trigonometry - H

Two boats starting at the same place and time, speed away along courses that form a 150º angle. If one boat travels at 54 miles per hour and the other boat travels at 30 miles per hour, determine how far apart the boats are after 20 minutes? A. 15.62 mi B. 18.43 mi C. 24.08 mi D. 27.13 mi E. 579.88 mi

Answer: D

Given a triangle with the following information provided: a = 38, b = 19, C = 122º, which of the following would we use first in order to solve the triangle: A. The Pythagorean Theorem B. The Law of Sines C. The Law of Cosines D. The Quadratic Formula

Answer: C




Trigonometry - H

Given the following information about a triangle: a = 29, b = 25, c = 10, and the measure of angle B = 93o Which one of the following is true? A. Side-lengths a, b, and c don’t satisfy the Triangle Inequality. B. We can use the Law of Sines to determine angles A or C. C. We cannot have a triangle since the longest side is not opposite the largest angle. D. The three side-lengths form a Pythagorean triple.

Given a triangle where a = 55, c = 80, and A = 35º, find two possible values for angle C.




Answers/Rubrics


Trigonometry - H

Given the following information about a triangle: a = 29, b = 25, c = 10, and the measure of angle B = 93o Which one of the following is true? A. Side-lengths a, b, and c don’t satisfy the Triangle Inequality. B. We can use the Law of Sines to determine angles A or C. C. We cannot have a triangle since the longest side is not opposite the largest angle. D. The three side-lengths form a Pythagorean triple.

Answer: C

Given a triangle where a = 55, c = 80, and A = 35º, find two possible values for angle C.

1

sin 35º sin55 80

80sin 35sin55

=56.54º,123.46º

C

C −

=

°⎛ ⎞= ⎜ ⎟⎝ ⎠

A 4-point answer clearly shows correct use of the Law of Sines to get correct values. A 3-point answer shows a correct solution for the 1st angle but makes a minor computation error in determining the second angle. A 2-point answer shows correct set-up and work to find only the first angle. A 1-point answer sets up the proper Law of Sines proportion but fails to get a single correct answer. A 0-point answer shows no understanding of the problem.




A ramp leading to a freeway overpass is 540 ft long and rises 38 ft. What is the average angle of inclination of the ramp to the nearest tenth of a degree? A. 0.07o B. 4.0o C. 33.3o D. 96.0o

To approximate the height of a radio tower, Mark counts off 72 feet from the base of the tower and then measures the angle of elevation from the ground to the top of the tower from that point to be 40o. Approximately how tall is the tower? A. 46.3 ft B. 55.2 ft C. 60.4 ft D. 72 ft

Trigonometry - I




Answers/Rubrics


Trigonometry - I

A ramp leading to a freeway overpass is 540 ft long and rises 38 ft. What is the average angle of inclination of the ramp to the nearest tenth of a degree? A. 0.07o B. 4.0o C. 33.3o D. 96.0o

Answer: B

To approximate the height of a radio tower, Mark counts off 72 feet from the base of the tower and then measures the angle of elevation from the ground to the top of the tower from that point to be 40o. Approximately how tall is the tower? A. 46.3 ft B. 55.2 ft C. 60.4 ft D. 72 ft

Answer: C




Trigonometry - I

A boat leaves harbor and travels at 30 knots on a bearing of 83o. After four hours, it changes course to a bearing of 138o and continues at 30 knots for three hours. After the entire seven-hour trip, how far is the boat from its starting point? A. 93.4 nautical miles B. 140.5 nautical miles C. 186.8 nautical miles D. 373.6 nautical miles

From the top of a 225 ft. building, a man observes a car moving towards the building. If the angle of depression from the man to the car changes from 18º to 39º during the period of observation, determine how far the car travels. Include a diagram to back up your work.




Answers/Rubrics

High Complexity

Trigonometry - I

A boat leaves harbor and travels at 30 knots on a bearing of 83o. After four hours, it changes course to a bearing of 138o and continues at 30 knots for three hours. After the entire seven-hour trip, how far is the boat from its starting point? A. 93.4 nautical miles B. 140.5 nautical miles C. 186.8 nautical miles D. 373.6 nautical miles

Answer: C




Answers/Rubrics


Trigonometry - I

From the top of a 225 ft. building, a man observes a car moving towards the building. If the angle of depression from the man to the car changes from 18º to 39º during the period of observation, determine how far the car travels. Include a diagram to back up your work. From the top of a 225 ft. building, a man observes a car moving towards the building. If the angle of depression from the man to the car changes from 18º to 39º during the period of observation, determine how far the car travels. Include a diagram to back up your work.

225 ft

72o

x

Time1

225 ft

51o

y

Time 2

225 ft

18o

21o39o

time 1time 2

tan 72° =x

225x = 225 tan 72° = 692.41 ft

tan51° =

y225

y = 225 ⋅ tan51° = 277.85 ft

note: you could also solve tan18° =225

x note: you could also solve tan 39° =

225y

Change in Distance = 692.41 ft – 277.85 ft = 414.56 ft. A 4-point answer shows both tangent problems set-up and solved properly and gets the correct difference. A 3-point answer gets both the setups and substitutions correct, but makes one mistake in the calculations somewhere along the way. A 2-point answer sets up the diagram(s) properly and shows the need to find a difference but is unable to use an appropriate trig ratio to determine these distances. A 1-point answer attempts to set up the diagram(s) properly so as to get the necessary angle measures, but is missing crucial pieces. A 0-point answer shows no mathematical understanding of the problem.




What is the factored form for the expression 1 – cos3x? A. (1 – cos x) (1 + cos x + cos2x) B. (1 – cos x)3

C. (1 – cos x) (sin x + cos2x) D. (1 – cos x) (1 – 2 cos x + cos 2x)

Find all solutions to the equation 2sin 2sin 1 0x x+ + = in the interval[0,2 ]π .

A. 3,2 2

x π π=

B. 3, ,2 2

x π ππ=

C. 2

x π=

D. 32

x π=

Trigonometry - J




Answers/Rubrics


Trigonometry - J

What is the factored form for the expression 1 – cos3x? A. (1 – cos x) (1 + cos x + cos2x) B. (1 – cos x)3

C. (1 – cos x) (sin x + cos2x) D. (1 – cos x) (1 – 2 cos x + cos 2x)

Answer: A

Find all solutions to the equation 2sin 2sin 1 0x x+ + = in the interval[0,2 ]π .

A. 3,2 2

x π π=

B. 3, ,2 2

x π ππ=

C. 2

x π=

D. 32

x π=

Answer: D




Trigonometry - J

Which values of t on [ 2 ,2 ]π π− satisfy 12cos t < − ?

A. 4 2 2 4, ,3 3 3 3π π π π⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∪

B. 4 2 2 4, ,3 3 3 3π π π π⎡ ⎤ ⎡ ⎤− −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∪

C. 4 4,3 3π π⎛ ⎞−⎜ ⎟

⎝ ⎠

D. 2 2,3 3π π⎡ ⎤−⎢ ⎥⎣ ⎦

Use a graphing calculator to find all solutions to the equation 1 costan2 1 cosx x

x+⎛ ⎞ =⎜ ⎟ −⎝ ⎠

in the

interval[0,2 ]π . Include a sketch of your graphs as well as an explanation of how you reached your answer.




Answers/Rubrics


Trigonometry - J

Which values of t on [ 2 ,2 ]π π− satisfy 12cos t < − ?

A. 4 2 2 4, ,3 3 3 3π π π π⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∪

B. 4 2 2 4, ,3 3 3 3π π π π⎡ ⎤ ⎡ ⎤− −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∪

C. 4 4,3 3π π⎛ ⎞−⎜ ⎟

⎝ ⎠

D. 2 2,3 3π π⎡ ⎤−⎢ ⎥⎣ ⎦

Answer: A

Use a graphing calculator to find all solutions to the equation 1 costan2 1 cosx x

x+⎛ ⎞ =⎜ ⎟ −⎝ ⎠

in the

interval[0,2 ]π . Include a sketch of your graphs as well as an explanation of how you reached your answer.

1.5708 3.14159 4.71239 6.28319

1(1.5708, 1.)

A 2-point response gives accurate sketches of the two graphs for the given interval and indicates the correct point where the two intersect. A 1-point response has one or more errors in one or both of the two graphs but uses otherwise valid reasoning to create an answer to the equation. A 0-point response shows no mathematical understanding of the problem.



Teacher Introduction Trigonometry

Periodic Functions

Periodic functions are functions where the values repeat themselves at regular intervals. There is a c such that f(t + c) = f(t) for every t in the functions domain, where c is the period length. The length can be found by determining the distance between two maxima (or 2 minima values). If the periodic function is shifted horizontally by its period, in either direction, the resulting graph will be the same as the original graph. The midline of a periodic function is a horizontal line halfway between the minimum and maximum values. The amplitude is the vertical distance, i.e. height difference, between the functions maximum and its midline. For example: F(x) = x – int(x) with Δx = 0.1 should be graphed by the student. The student should look not only at the graph, but at the table of values. This function is periodic with a period of 1.

Sine and Cosine Functions The sine and cosine functions should be studied using a unit circle with a radius of 1 and centered at the origin. Each point on the circle can be located by its angle of rotation, θ, where y = sin θ and x = cos θ. The pictures of these are included in the strategies for Learning Goal B. To form triangles, you extend a ray from the center to the side of the circle and draw an altitude to the x-axis. The altitude and x-axis form a right angle and the ray becomes the hypotenuse of the right triangle, with length = 1.

The angle θ is the angle the ray makes with the x-axis. The adjacent side is on the x-axis with length equal to the value of the x-coordinate. The opposite side is the altitude with length equal to the value of the y-coordinate. This can be generalized for non-unit circles of radius r: x = rcosθ and y = rsinθ. If θ is the angle measured from the positive x-axis, and P(x, y) is the point on the circle that intercepts the terminal ray, then a right triangle is formed with the hypotenuse as part of the

Unit Circle

(0, 1)

(0, -1)

(1, 0) (1, 0) θ

(x, y)



terminal ray. By using the Pythagorean Theorem, cos2θ + sin2θ = 1 (Pythagorean Identity) can be proved. On the unit circle, we represent angles as rotations of a ray counter-clockwise from the positive x-axis. The x-axis is the initial side of the angle and the ray is the terminal side of the angle. An angle formed in this way is said to be in standard position. Example: Use the height of a Ferris wheel by generating the points for 0o to 360o for the equation y = 225 + 225sinx and creating a scatterplot to illustrate this concept.

Radians 1 radian is the angle at the center of a unit circle which spans an arc of length one. Radians are commonly used in analytical trigonometry and in calculus. The formula for Arc length is: S = rθ . π radians = 180o Students will need to be familiar with changing the mode on the graphing calculator from degrees to radians and vice versa.

Graphs of Sine and Cosine Relate the unit circle to the graphs of the sine and cosine functions with special emphasis on the 2nd, 3rd, and 4th quadrants. When 0° < θ < 90°; cos θ, sin θ, and tan θ have positive values because x and y are both positive in the first quadrant. When 90° < θ < 180°; cos θ and tan θ have negative values because x is negative in the second quadrant. Sin θ has a positive value because y is positive in the second quadrant.

(0, 1)

(0, -1)

(1, 0) (-1, 0) θ

Initial Side

Terminal Side



When 180° < θ < 270°; cos θ and sin θ have negative values because x and y are negative in the third quadrant. Tan θ has a positive value. When 270° < θ < 360°; sin θ and tan θ have negative values because y is negative in the fourth quadrant. Cos θ has a positive value because x is positive in the fourth quadrant.

If students have trouble remembering the exact values of the sine and cosine functions for 6π ,

4π , and

3π , remind them of the relationships in a 30, 60, 90 and a 45, 45, 90 triangle.

3π

4π

2 2 1 1

6π

4π

3 1

As students compare the two graphs, they should recognize that both have a period of 2π, an amplitude of 1, a domain of (-∞, ∞) and a range of [-1, 1]. They should also note that the sine function is odd while the cosine function is an even. Like other functions, sine and cosine functions can be shifted horizontally, vertically, inverted, compressed, stretched, or a combination of those shifts.

Sinusoidal Functions

Any transformation of the sine and cosine function is called a sinusoidal function. Students will be working from the general form sin ( )y a b x c d= + + where the different variables have the following properties (note this is slightly different than what the book does): a is called the amplitude; it is the distance from the highest point of the sine curve to the

mid-line.

b does not have a name itself but is used to determine the period: 2pbπ= . The period is

how many times the graph completes a cycle in a 2π interval. d is the vertical shift; it tells how far up or down the entire sine curve is shifted. c gives the phase shift; how far left or right the graph is shifted (a positive c means shift

to the left; a negative c means shift to the right). One of the big challenges in analyzing sinusoidal functions is to turn the given function into something that looks like the above form so it can be analyzed easily. For example, the given equation 4 2sin(3 )y x π= + − is usually approached by factoring the 3 out, to give something

30o

60o 45o

45o



like ( )34 2sin 3y x π= + − , which allows us to see that the period length is 23π and the phase shift

is 3π units to the right.

One thing that will help when students learn to convert equations into this standard form is that they can put both into their graphing calculator. If they are correct, both graphs will wind up on top of each other.

Solving Trigonometric Equations

Remind students that it is better to remember the Pythagorean Theorem as leg2 + leg2 = hypotenuse2 rather than a2 + b2 = c2, since there is no guarantee that c is always the hypotenuse. There are two special right triangles. The first is a 45-45-90 triangle. The special ratio is 1:1: 2 . The second is a 30-60-90 triangle. The special ratio is 1: 3 : 2. Trigonometry is based on similar right triangles. The sine (sin) of an angle is the ratio of the opposite side to the hypotenuse. The cosine (cos) of an angle is the ratio of the adjacent side to the hypotenuse. The cosine of an angle is the ratio of the adjacent side to the hypotenuse. The tangent (tan) of an angle is the ratio of the opposite side to the adjacent side. There are many different ways to help your students remember the sine, cosine, and tangent functions.

• Use the old Indian chief SOH CAH TOA. Tell a story of how a great Indian chief was also a great mathematician. And he developed sine, cosine, and tangent to match his name.

SOH (sin = opp / hyp) CAH (cos = adj / hyp) TOA (tan = opp / adj)

• The following phrase could also be used.

Some Old Horse Caught Another Horse Taking Oats Away

The angle of elevation is the angle between the line of sight and the horizontal when looking up. The angle of depression is the angle between the line of sight and the horizontal when looking down. It is helpful to remember that the angle of elevation and the angle of depression are alternate interior angles to each other. Real life applications are architecture and engineering.

depression

elevation

Look down to person

Look up to bird



Non-Right-Angle Trigonometry (Law of Sines/Law of Cosines) We use the Law of Sines and Law of Cosines when we need to solve a triangle that is not a right triangle. By solving, we mean determining the missing side lengths and angle measures, given a minimum of three of these. Students are already familiar with the Pythagorean Theorem which relates the three side-lengths of a right triangle to each other. These two new laws are also about relating information to other information. To use either law, we must use the convention that the three sides of a triangle are labeled a, b, and c, and that their opposite angles are labeled A, B, and C, respectively. The Law of Sines states that for any triangle:

sin sin sinA B Ca b c

= =

What we need to know in order to use this law is one side length and its opposite angle. In other words, we must know either A and a, B and b, or C and c. The other piece of information we are given determines which other fraction we will pick out. For example, if we are given the values

of C, c, and B, then we will set up the proportion, sin sinB Cb c

= , allowing us to find the length of

side b. We cannot use the Law of Sines if we do not have three pieces of information, and they must meet the requirement we just mentioned. In this last example, if we were given C, c, and b, then we would use the same proportion, but that would leave us with a value for the sin of B, and we would still have to take the inverse sine in order to determine the angle. There are a few situations where the Law of Sines yields unexpected results. First, consider a case where a = 7, b = 8, and A = 100º. Attempting to use the Law of Sines to find the measure of B will give no answer (try it yourself). This is exactly as we expect and want, since in any triangle, the longest side must lie opposite the biggest angle. This cannot be, because there cannot exist any angle bigger than 100º in this triangle, but the 100º is opposite the 7 which isn’t the longest side. Another interesting type of result is what is called the ambiguous case. Here, the Law of Sines may give two possible (and viable) results for a missing angle. The Law of Cosines reads: 2 2 2 2 cosc a b ab C= + − This law relates the three side-lengths to the measure of one angle. The two most common uses of this law are: 1. To find the missing side when the given information is arranged in side angle side formation. 2. To find a missing angle when all three side-lengths are known. Here, what we are really doing is solving the above equation as if it were in this form:

2 2 2

1cos2

c a bCab

− ⎛ ⎞− −= ⎜ ⎟−⎝ ⎠



One important property of the Law of Cosines that should be pointed out to students is that when angle C is 90º (when we have a right triangle), then the equation degenerates into the familiar Pythagorean Theorem. In solving a triangle, you will not have to use the Law of Cosines more than once. Once you know all three side-lengths and an angle, you can use the Law of Sines to determine another angle. Also, don’t forget such geometry basics as the fact that the angle measures of a triangle must add up to 180º. If you have 2 of them, it’s a simple computation to get the third.

Trigonometric Identities

The section on trigonometric identities should emphasize their use in verifying trigonometric formulas, such as the double angle formula and the reductions rules, and their application in solving trigonometric equations. Although students should have some experience in verifying identities algebraically, this is not the focus of the section. Students should know that, although identities can be verified graphically, this does not constitute a proof.

Trigonometric Regression Trigonometric Regression is expected to be done using technology. Please refer to your Graphing Calculator Resource Manual.



TEACHING STRATEGIES/ACTIVITIES

Core: Learning Goal A: Uses radian and degree angle measure to solve problems and perform conversions as needed. 1. A blank “Unit Circle” (included in this Curriculum Guide) and a “Circle With Radian

Measure” (included in this Curriculum Guide) have been provided for the teacher to use at his/her discretion.

2. “The Radian Snow.....” (included in this Curriculum Guide) can also be used to help students learn the radian measurements. Students may turn it into a snowman, a snowcat, etc.

Learning Goal B: Uses the unit circle to explain the circular properties and periodic nature of trigonometric functions and to find the trigonometric ratios of any angle. 1. Use the Unit Circle Investigation (included in this Curriculum Guide) to help the students

discover the relationships of the unit circle. Provide students with the handout, scissors, tracing paper, and the unit circle.

2. As a way to reinforce the learning of the angle measures in radians the corresponding coordinates on the unit circle, the students will play the game, “The Radian Walk” (included in this Curriculum Guide).

3. A “Circle With Sine and Cosine Coordinates” (included in this Curriculum Guide) has been provided for the teacher to use at his/her discretion.

Learning Goal C: Describes and compares the characteristics of the trigonometric functions (with and without the use of technology) for sine, cosine, tangent, cotangent, cosecant, and secant. 1. The “Sine Cosine Game” (included in this Curriculum Guide) can be used to reinforce the

cyclic nature of the trigonometric function. There are four different games which increase in level of difficulty.

2. As students increase their mathematics learning level, they often learn that Mathematics is not the dreaded subject that they have previously encountered. “The Story of Joe S____.” (included in this Curriculum Guide) is a play on mathematical words which students may find enjoyable.

Vocabulary: Mathematical model, domain, range, function, continuous, jump discontinuity, increasing, decreasing, constant, lower bound, upper bound, boundedness, local extrema, absolute extrema, odd function, even function, asymptote, sine function, cosine function, tangent function, secant function, cosecant function, inverse sine function, inverse cosine function, inverse tangent function, inverse, relation, transformation, translation, reflection, stretch, shrink, regression, angle, central angle, degree, minute, second, bearing, radian, arc length, sector, similar, standard position, sine, cosine, tangent, cosecant, secant, cotangent, solving a triangle, right triangle, vertex, positive angle, negative angle, terminal ray, initial ray, standard position, coterminal angles, reference angle, quadrantal angle, periodic function, period, sinusoid, amplitude, frequency, phase shift, vertical shift, damped oscillation, damping factor, angle of elevation, angle of depression, simple harmonic motion, trigonometric identity, Pythagorean identities, trigonometric equation, double-angle identity, half-angle identity, Law of Sines, Law of Cosines, semiperimeter, Heron’s Formula.



Learning Goal D: Determines solutions to trigonometric equations. 1. Student will use their knowledge of unit circle as a way to solve simple trigonometric

equations both graphically and with technology in “Solving Trigonometric Equations” (included in this Curriculum Guide).

2. Have students complete the activity “Problem Solving: Trigonometric Ratios” (included in Curriculum Guide) to apply their knowledge of trigonometric ratios.

Learning Goal E: Describes how a change in the value of any constant in a general-form trigonometric equation such as y = a sin (b-x) + c affects the graph of the equation. 1. Reinforce the properties of the unit circle and introduce students to periodic functions using

the “Pasta Waves” activity (included in this Curriculum Guide). Students will need the handout “Pasta Waves”, a copy of the “Unit Circle” (included in this Curriculum Guide), a copy of the “S(x) Graph” (included in this Curriculum Guide), a copy of the “C(x) Graph” (included in this Curriculum Guide), a copy of the “Observations and Predictions” sheet (included in this Curriculum Guide), spaghetti, and glue.

2. In “Getting in Shape” (included in this Curriculum Guide), the students will explore the concept of amplitude.

3. “Speeding Up” (included in this Curriculum Guide), is an extension to “Getting in Shape” which deals with changing the period of a function.

4. The third activity in this series is “Running With a Friend” (included in the Curriculum Guide. This activity addresses the period of a function.

5. Students will used the “Sine Curve Equation” (included in this Curriculum Guide) to show their understanding of the amplitude, period, and phase shift of the sine curve.

Learning Goal F: Represents the inverse of a trigonometric function symbolically and graphically 1. Once the student understands the unit circle and an inverse function, they should complete

“Finding the Inverse Without a Calculator” (included in this Curriculum Guide). 2. In “Match the Inverse” (included in this Curriculum Guide), the student will match the graph

of the inverse with its equation. Learning Goal G: Creates a scatterplot of bivariate data, identify a trigonometric function to model the data., and use that model to identify patterns and make predictions. 1. In “Daylight Hours” (included in this Curriculum Guide), students will explore the use of

real life data to create a trigonometric equation to model the data. Learning Goal H: Derives and applies the basic trigonometric identities; i.e. angle addition, angle subtraction, and double-angle. 1. Students will play the game “Tic-Tac Trig – 4 in a Row” (included in this Curriculum Guide)

as a way to solidify their understanding of the simplification of the multiplication of trigonometric functions.

Learning Goal I: Uses trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines and Law of Cosines.



1. In the “Sine Spaghetti Investigation” (included in this Curriculum Guide), students will investigate the Law of Sines.

2. Use the “Law of Sines: Real Life Applications” (included in this Curriculum Guide) to show some applications of the Law of Sines.

3. “Sines Just Aren’t Enough” will give the students is using the Law of Cosines when the Law of Sines can not be used.

Learning Goal J: Models and solve problems using trigonometry. 1. The Textbook contains a wide selection of problems throughout this entire section. In “A

Variety of Problems” (included in this Curriculum Guide), the students will use various trigonometric functions and properties to find the solutions.

2. “The Big Wheel” (included in this Curriculum Guide) is another application problem that the students should solve.

Reteach: 1. Have students practice finding the missing side or angle of a right triangle with the “Find the

Missing Side or Angle” activity (included in this Curriculum Guide). Students will need a copy of the activity and a scientific or graphing calculator.

2. Have students create their own unit circle to have as a reference with the “Simple Circle” activity (included in this Curriculum Guide) or the “Paper Plate” activity (included in this Curriculum Guide).

3. Students practice locating the special right triangles on the unit circle by completing the “Crop Circles” activity (included in this Curriculum Guide). Have students discuss with a small group or the whole class their strategies for determining how much of each circle to shade.

4. Have students complete the activity “Memory Match” (included in this Curriculum Guide) to reinforce right triangle terminology. Students can work in groups of three or four. First place all cards facedown and then have each student take turns drawing two cards. If the two cards drawn go together as a pair the student will keep it as a match. Students take turns drawing. The student with the most pairs or matches wins. Students will need a scientific calculator.

Extension: 1. “When Good Trig Goes Bad” requires the student to use many of the ideas to prove

trigonometric ideas.

RESOURCES Learning Goal A: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 351-371 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp. 77-80 Learning Goal B: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 372-385 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp. 81-82



Learning Goal C: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 372-385 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp. 81-82 Learning Goal D: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 426-437 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp. 91-92 Learning Goal E: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 386-415 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp. 83-88 Learning Goal F: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 416-425 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp. 89-90 Learning Goal G: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 426-442 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp. 91-92 Learning Goal H: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 443-477 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp. 99-106 Learning Goal I: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 478-496 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp. 107-110 Learning Goal J: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 351-496 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp. 77-115



Unit Circle

Name

Trigonometry - A



Circle With Radian Measure

Name

2π

23π

3π

34π

4π

56π

6π

π 0 2π

76π 11

6π

54π 7

4π

43π 5

3π

32π

Trigonometry - A



The Radian Snow.....

Name The snow..... picture has a unit circle for its base. You are to label the sixteen points on the unit circle with the radian measure inside the circle and the coordinates of the points outside the circle. Then color and decorate the snow.....

Trigonometry - A



The Radian Snow..... Answer Key

These are some examples of student work.

Trigonometry - A



Unit Circle Investigation

Name Assume that the radius of the outlined circle is 1 unit. 1. Label the ordered pairs on the x and y-axes. Draw a perpendicular line segment from the

point on the outlined circle where the π6

ray crosses the outlined circle. Trace the right triangle

defined by the x-axis, the perpendicular segment, and the π6

ray. Cut it out. Label the lengths of

the sides and the angles of the triangle.

2. Make another triangle by dropping a perpendicular from the point where the π4

ray crosses

the circle. Place the 30º-60º triangle on its outline. Label the ordered pair where the vertex

crosses the circle. What is the cos π6

(30º)? What is the sin π6

(30º)? How do you know?

3. Align the triangle between the 60º angle and the x-axis. Label the ordered pair where the

vertex crosses the circle. What is the cos π3


(60º)? How are these

related to the sine and cosine of π6

(30º). Explain this relationship in terms of the right triangles.

4. Align the right triangle along the negative x-axis under the 2π3

ray. Label the ordered pair

where the ray crosses the circle. Move the triangle so that it is aligned under the 5π6

ray. Label

the ordered pair where the ray crosses the circle. These triangles are called the reference triangles and allow you to find the trigonometric function of angles outside the first quadrant. What are the sine and cosine of these angles? 5. How are the sine and cosine of angles in the second quadrant related to the angles in the first quadrant? 6. Align the triangle with the x-axis in the 3rd and 4th quadrants and find the sine and cosine of each of the related angles in these quadrants.

Trigonometry - B



7. Find the angles and ordered pairs associated with π4

around the circle.

8. Find the sine and cosine of π12

from the calculator. Give the degree measure of that angle.

Mark all the angles whose sines and cosines can be determined from this. Give their measures in degrees and radians. Give the sine and cosine of each. Check your answers with the calculator. 9. Pick another angle θ between 0º and 45 º. Use your calculator to evaluate sin θ and cos θ. On your circle, draw in the arc that approximates this measure. Give the measures of all the angles whose sines and cosines can be determined from this, and give their sines and cosines. Check your answers with the calculator. 10. In what quadrants are sine and cosine positive and negative?

Trigonometry - B



11. Suppose you know the measure of an angle θ and 0< θ <45º. Draw in all the other angles on the circle whose sine and cosine can be determined from the cos θ and the sin θ. Write expressions for each of these angles in terms of θ. Write the ordered pairs for each of these angles in terms of cos θ and sin θ.

(cos θ, sin θ) θ

Trigonometry - B



Unit Circle

Name

Trigonometry - B



Unit Circle Investigation Answer Key

Assume that the radius of the outlined circle is 1 unit. 1. Label the ordered pairs on the x and y-axes. Draw a perpendicular line segment from the

point on the outlined circle where the π6

ray crosses the outlined circle. Trace the right triangle

defined by the x-axis, the perpendicular segment, and the π6

ray. Cut it out. Label the lengths of

the sides and the angles of the triangle.

2. Make another triangle by dropping a perpendicular from the point where the π4

ray crosses

the circle. Place the 30º-60º triangle on its outline. Label the ordered pair where the vertex

crosses the circle. What is the cos π6


(30º)? How do you know?

cos 6π = 3

2 and sin

6π = 1

2

They are the x and y coordinates of the terminal ray. 3. Align the triangle between the 60º angle and the x-axis. Label the ordered pair where the

vertex crosses the circle. What is the cos π3


(60º)? How are these

related to the sine and cosine of π6

(30º). Explain this relationship in terms of the right triangles.

cos 13 2π = and sin 3

3 2π =

The values of the functions are interchanged. This relationship can be seen in terms of a 30-60-90 triangle.

4. Align the right triangle along the negative x-axis under the 2π3

ray. Label the ordered pair

where the ray crosses the circle. Move the triangle so that it is aligned under the 5π6

ray. Label

the ordered pair where the ray crosses the circle. These triangles are called the reference triangles and allow you to find the trigonometric function of angles outside the first quadrant. What are the sine and cosine of these angles?

sin 2 33 2π = , cos 2 1

3 2π = − , sin 5

6π = 1

2, cos 5

6π = 3

2−

5. How are the sine and cosine of angles in the second quadrant related to the angles in the first quadrant? Sin values are the same as the ones in the first quadrant. Cosine values are the opposite of the ones in the first quadrant. 6. Align the triangle with the x-axis in the 3rd and 4th quadrants and find the sine and cosine of each of the related angles in these quadrants. See student answers on the unit circle.

Trigonometry - B



7. Find the angles and ordered pairs associated with π4

around the circle.

See student answers on the unit circle.

8. Find the sine and cosine of π12

from the calculator. Give the degree measure of that angle.

Mark all the angles whose sines and cosines can be determined from this. Give their measures in degrees and radians. Give the sine and cosine of each. Check your answers with the calculator.

12π = 15o, 5

12π = 75o, 7

12π = 105o, 11

12π = 165o, 13

12π = 195o, 17

12π = 255o, 19

12π = 285o, 23

12π = 345o

sin12π = 0.2588, cos

12π = 0.9659, sin 5

12π = 0.9659, cos 5

12π = 0.2588, sin 7

12π =0.9659

cos 712π = -0.2588, sin 11

12π = 0.2588, cos 11

12π = -0.9659, sin 13

12π = -0.2588, cos 13

12π = -0.9659

sin 1712

π = -0.9659, cos 1712

π = -0.2588, sin 1912

π = -0.9659, cos 1912

π = 0.2588, sin 2312

π = -0.2588

cos 2312

π = 0.9659

9. Pick another angle θ between 0º and 45 º. Use your calculator to evaluate sin θ and cos θ. On your circle, draw in the arc that approximates this measure. Give the measures of all the angles whose sines and cosines can be determined from this, and give their sines and cosines. Check your answers with the calculator. Student answers will vary based on the value of θ that they choose. Check the students unit circle. 10. In what quadrants are sine and cosine positive and negative? Sine is positive in the first and second quadrants. Sine is negative in the third and fourth quadrants. Cosine is positive in the first and fourth quadrants. Cosine is negative in the second and third quadrants.

Trigonometry - B



11. Suppose you know the measure of an angle θ and 0< θ <45º. Draw in all the other angles on the circle whose sine and cosine can be determined from the cos θ and the sin θ. Write expressions for each of these angles in terms of θ. Write the ordered pairs for each of these angles in terms of cos θ and sin θ. (-sinθ, cosθ) (sin θ, cos θ) (-cos θ,sin θ) (-cos θ, -sin θ) (cos θ, -sin θ) (-sin θ, -cos θ) (sin θ, -cos θ)

(cos θ, sin θ) θ

Trigonometry - B



The Radian Walk

The teacher will make a unit circle with diameter about 12 feet on the floor in the middle of the classroom. Make the x- and y-axes to mark the 90 degree angles. Label them so that students know the location of 0 radians. Mark the 30 degree, 45 degree, and 60 degree angles in each quadrant.

Place a spinner at the origin. Turn on a tape or CD player and have the students walk around the circle until the music stops. At that point each student must be on one of the marked angles of the unit circle. Spin the spinner. The spinner indicates the student who must name her or his coordinates and place on the unit circle.

If that student makes a mistake, she or he is eliminated from the game. The person remaining on the circle after all others have dropped out is the winner. If your class size is larger than fifteen or sixteen students, you may want to use two unit circles.

Trigonometry - B



Circle With Sine and Cosine Coordinates

Name (0, 1)

1 3,2 2

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

1 3,2 2

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

2 2,2 2

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

2 2,2 2

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

3 1,2 2

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

3 1,2 2

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

(-1, 0) (1, 0)

3 1,2 2

⎛ ⎞− −⎜ ⎟⎜ ⎟⎝ ⎠

3 1,2 2

⎛ ⎞−⎜ ⎟⎜ ⎟

⎝ ⎠

2 2,2 2

⎛ ⎞− −⎜ ⎟⎜ ⎟⎝ ⎠

2 2,2 2

⎛ ⎞−⎜ ⎟⎜ ⎟

⎝ ⎠

1 3,2 2

⎛ ⎞− −⎜ ⎟⎜ ⎟⎝ ⎠

1 3,2 2

⎛ ⎞−⎜ ⎟⎜ ⎟

⎝ ⎠

(0, -1)

Trigonometry - B



Sine Cosine Game

After students have been introduced to the sine and cosine functions, begin a class period with this game. Divide the class into pairs and distribute single sheets of paper to each pair. On the sheet are two columns. The left-hand column contains expression involving either the sine or cosine function. In the right-hand column are actual decimal approximations of the expression in the left-hand column, but in a random order. The values are chosen such that only one of the five decimal values is plausible for each function. At the signal from the teacher, the pairs of students have 3 minutes to match each expression in the left-hand column with its proper decimal representation in the right-hand column. Calculators are not permitted. Any pair is a winner if they obtain all five correct matches. After the introduction of all six trigonometric functions, the game can be played with mixing the inverse functions or mixing all of the functions. Four different matching games are provided for your use.

Trigonometry - C



Sine Cosine Game

Names: Game # 1 _______ cos 35 A. -0.9961 _______ cos 98 B. 0.8192 _______ cos 4 C. 0.0349 _______ cos 175 D. -0.1391 _______ cos 272 E. 0.9976

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Names: Game # 2 _______ sin 224 A. 0.1908 _______ sin 52 B. 0.7888 _______ sin 290 C. -0.0871 _______ sin 355 D. -0.6946 _______ sin 11 E. -0.9396

Trigonometry - C



Names: Game # 3 _______ cos 7 A. -1.2360 _______ sec 98 B. 1.0456 _______ cos 304 C. 0.5591 _______ sec 17 D. -7.1852 _______ sec 144 E. 0.9925

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Names: Game # 4 _______ csc 130 A. -28.653 _______ sin 289 B. 1.3054 _______ sin 185 C. -1.0402 _______ csc 254 D. -0.9455 _______ csc 358 E. -0.0871

Trigonometry - C



Sine Cosine Game Answer Key

Names: Game # 1 __B____ cos 35 A. -0.9961 __D____ cos 98 B. 0.8192 __E____ cos 4 C. 0.0349 __A____ cos 175 D. -0.1391 __C____ cos 272 E. 0.9976

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Names: Game # 2 __D____ sin 224 A. 0.1908 __B____ sin 52 B. 0.7888 __E____ sin 290 C. -0.0871 __C____ sin 355 D. -0.6946 __A____ sin 11 E. -0.9396

Trigonometry - C



Names: Game # 3 __E____ cos 7 A. -1.2360 __D____ sec 98 B. 1.0456 __C____ cos 304 C. 0.5591 __B____ sec 17 D. -7.1852 __A____ sec 144 E. 0.9925

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Names: Game # 4 __B____ csc 130 A. -28.653 __D____ sin 289 B. 1.3054 __E____ sin 185 C. -1.0402 __C____ csc 254 D. -0.9455 __A____ csc 358 E. -0.0871

Trigonometry - C



The Story of Joe S _ _ _ .

Name Fill in the following blanks with letters to make math words. One day Joe S _ _ _ went to call on his new neighbors who lived in the a _ _ _ _ _ _ _ house.

He was a handsome t _ _ _ _ _ _ , a confirmed bachelor. Joe was met at the door by two

sisters who had anything but c _ _ _ _ _ _ _ _ f _ _ _ _ _ _ . The first, Deca Gon, had real

c _ _ _ _ _ _ _ _ _ _ _ problems. Her d _ _ _ _ _ _ _ _ _ _ _ _ c _ _ _ _ _ were intersected at

various a _ _ _ _ _ by p _ _ _ _ _ _ _ l _ _ _ _ . The second sister, Polly Gon, was dressed

in a pretty c _ _ _ _ _ _ _ _ _ s _ _ and it was obvious that her natural c _ _ _ _ _ ran into

i _ _ _ _ _ _ _ _ n _ _ _ _ _ _ . Just looking from the first to the second, Joe found his

i _ _ _ _ _ _ _ c _ _ _ _ _ _ _ _ _ _ rapidly. What poor Joe did not know was that Polly knew

all the a _ _ _ _ _ and was an expert at taking s _ _ _ _ _ _ . Joe was invited to come in and sit

down. Deca proved to be as s _ _ _ _ _ as she looked and just sat there like a l _ _ . Polly, at

a given s _ _ _ , sent Deca out to find some r _ _ _ _ to make tea. While she was gone, Polly

served Joe s _ _ p _ . Then she used the c _ _ _ _ _ _ _ _ _ _ _ _ a _ _ _ _ and Joe was

soon r _ _ _ _ _ _ to z _ _ _ p _ _ _ _ . Next, she introduced an "I _ ... T _ _ _

p _ _ _ _ _ _ _ _ _ _ ." That is, if Joe would marry her, then ... Well, Joe said, "yes," but then

began to consider the possibility of spending the rest of his life a _ _ _ _ _ _ _ to her s _ _ _ .

He began to get very nervous and his head was going in c _ _ _ _ _ _ . He knew she was a

c _ _ _ _ _ _ n _ _ _ _ _ and he was really afraid of her. So, he tried to make her think that he

was o _ _ by asking, "Do u _ _ _ ?", pretending that he did. When that didn't work, he stood

up and announced that he was going to the bus t _ _ _ _ _ _ _ to m _ _ _ _ _ , his true

beloved.

Trigonometry - C



The Story of Joe Sine. Answer Key

Fill in the following blanks with letters to make math words. One day Joe S ine went to call on his new neighbors who lived in the adjacent house.

He was a handsome tangent , a confirmed bachelor. Joe was met at the door by two

sisters who had anything but congruent figures . The first, Deca Gon, had real

construction problems. Her discontinuous curves were intersected at

various angles by parallel lines . The second sister, Polly Gon, was dressed

in a pretty coordinate set and it was obvious that her natural curves ran into

imaginary numbers . Just looking from the first to the second, Joe found his

interest compounding rapidly. What poor Joe did not know was that Polly knew

all the angles and was an expert at taking squares. Joe was invited to come in and sit

down. Deca proved to be as square as she looked and just sat there like a log . Polly, at

a given sign , sent Deca out to find some roots to make tea. While she was gone, Polly

served Joe sum pi . Then she used the corresponding angle and Joe was

soon reduced to zero power . Next, she introduced an "If ... Then

proposition ." That is, if Joe would marry her, then ... Well, Joe said, "yes," but then

began to consider the possibility of spending the rest of his life adjacent to her side .

He began to get very nervous and his head was going in circles . He knew she was a

complex number and he was really afraid of her. So, he tried to make her think that he

was odd by asking, "Do unit ?", pretending that he did. When that didn't work, he stood

up and announced that he was going to the bus terminal to median , his true beloved.

Trigonometry - C



Solving Trigonometric Equations

Name 1. Use the unit circle to find all the solutions to the following equations in the interval [0,2π].

A. sin x = 32

B. cos x = − 22

C. tan x = − 3 D. csc x = −2 E. cos x = 0

Solve each of the equations graphically. Sketch your graphs below. All of these equations have more than one solution. Explain why in terms of the unit circle and in terms of the graph.

Trigonometry - D



2. Use your graphing calculator to find all the solutions to the following equations in the interval [0,2π]. Sketch your graphs below.

A. sin2x = 32

B. cos4x = − 22

C. tan2x = − 3 D. csc 3x = −2 E. cos6x = 0

3. For each equation, give the number of solutions and explain the number of solutions in terms of the period of the trigonometric function.

A. sin2x = 32

B. cos4x = − 22

C. tan2x = − 3 D. csc3x = −2 E. cos6x = 0 4. Give the period of the trigonometric function and then predict the number of solutions of each equation on the interval [0,2π]. Find the solutions. A. sin4 x = −.9903 B. tan.25x = 4

Trigonometry - D



Solving Trigonometric Equations Answer Key

1. Use the unit circle to find all the solutions to the following equations in the interval [0,2π].

A. sin x = 32

B. cos x = − 22

C. tan x = − 3 D. csc x = −2 E. cos x = 0

Solve each of the equations graphically. Sketch your graphs below. All of these equations have more than one solution. Explain why in terms of the unit circle and in terms of the graph. The trigonometric functions are each positive in two quadrants and negative in two quadrants. In terms of the graph, the graph is divided into halves, the part that is positive and above the x axis and the part that is negative and below the x axis.

1.5708 3.14159 4.71239 6.28319

-1

1

y = sin(x)

3π 2

3π

1.5708 3.14159 4.71239 6.28319

-1

1

y = cos(x)

34π 5

4π

1.5708 3.14159 4.71239 6.28319

-2

-1

1

2y = tan (x)

23π

53π

1.5708 3.14159 4.71239 6.28319

-3

-2

-1

1

2

3y = csc (x)

116π

76π

1.5708 3.14159 4.71239 6.28319

-1

1

y = cos (x)

2π 3

2π

Trigonometry - D



1.5708 3.14159 4.71239 6.28319

-1

1 (1.0472, .866025)

(.523599, .866025) (3.66519, .866025)

(4.18879, .866025)

A

1.5708 3.14159 4.71239 6.28319

-1

1

B

(2.553, -.707)

(2.160, -.707)

(.982, -.707)(.589, -.707)

(4.123, -.707)

(3.731, -.707)(5.301, -.707)

(5.694, -.707)

1.5708 3.14159 4.71239 6.28319

-2

-1

1

2C

(1.0472, -1.73205)

(2.61799, -1.73205)

(4.18879, -1.73205)

(5.75959, -1.73205)

1.5708 3.14159 4.71239 6.28319

-3

-2

-1

1

2D

(1.22173, -2.)

(1.91986, -2.)

(3.31613, -2.)

(4.01426, -2.)

(5.41052, -2.)

(6.10865, -2.)

1.5708 3.14159 4.71239 6.28319

-1

1

E

1.5708 3.14159 4.71239 6.28319

-1

1

1.5708 3.14159 4.71239 6.28319

-1

1

2

3

4

5

(5.30327, 4.)

2. Use your graphing calculator to find all the solutions to the following equations in the interval [0,2π]. Sketch your graphs below.

A. sin2x = 32

B. cos4x = − 22

C. tan2x = − 3 D. csc3x = −2 E. cos6x = 0

3. For each equation, give the number of solutions and explain the number of solutions in terms of the period of the trigonometric function.

A. sin2x = 32

B. cos4x = − 22

4 solutions, 2 periods 8 solutions, 4 periods

C. tan2x = − 3 D. csc3x = −2 E. cos6x = 0 4 solutions, 2 periods 6 solutions, 3 periods 12 solutions, 6 periods The number of periods is equal to the coefficient in front of the variable. The number of solutions is twice the number of periods. 4. Give the period of the trigonometric function and then predict the number of solutions of each equation on the interval [0,2π]. Find the solutions. A. sin4 x = −.9903 B. tan.25x = 4

Period is 2π , there will be 8 solutions. Period is 8π, there will be 1 solution in the given

interval.

(1.14325, -.9903),(1.21295, -.9903),(2.71404, -.9903),(2.78374, -.9903) (4.28484, -.9903),(4.35454, -.9903),(5.85564, -.9903),(5.92534, -.9903)

Trigonometry - D



B (0,5)

(3,4)

(5,0)A C θ

Problem Solving: Trigonometric Ratios Name ____________________________ Materials: scientific calculator

1. Use the information given in the figure below to determine the sine, cosine, and tangent of

∠θ. Explain your answer. Sin θ = _______ Cos θ = _______ Tan θ = _______ 2. Use the information given in the figure below to determine the perimeter of rectangle ABCD.

Support your answer by showing your work. Perimeter = ____________ 3. Which of the following trigonometric ratios: sine, cosine, or tangent of an angle can have a value greater than 1? Why is it that the values of the other two trigonometric ratios can never be greater than 1? Explain.

100 cm 35°

B C

A D

Trigonometry - D



4. John, an employee of the U.S. Forestry Service has been asked to determine the height of a

tall tree in Wayne National Forest. He uses an angle measuring device to determine the angle of elevation (angle formed by the line of sight to the top of the tree and a horizontal) to be about 33°. He walks off 40 paces to the base of the tree. If each pace is .6 meters, how tall is the tree to the nearest meter? Support your answer by showing your work and including a diagram.

5. Determine the perimeter to the nearest centimeter and the area to the nearest square

centimeter of the triangle shown below. Support your answer by showing your work and giving an explanation.

6. Use what you know about the side lengths of special right triangles to complete the following

table. Express your answers in simplified radical form.

30° 45° 60°

Sin

Cos

Tan

10 cm

A C

B

45°

45°

60°

30°

Trigonometry - D



B (0,5)

(3,4)

(5,0)A C θ

Problem Solving: Trigonometric Ratios Answer Key

1. Use the information given in the figure below to determine the sine, cosine, and tangent of

∠θ. Explain your answer. Sin θ = 45

Cos θ = 35

Tan θ = 43

Solution: The lengths of AC and BC can be determined by using the coordinates of point B(3,4). The length of AB can be determined by using the fact that it is a radius of a circle. AB = 5, BC = 4, and AC = 3. By definition:

BC 4 AC 3sin θ = = ; cos θ = = ; andAB 5 AB 5BC 4tan θ = =AC 3

2. Use the information given in the figure below to determine the perimeter of rectangle ABCD.

Support your answer by showing your work. Solution: By definition:

BC BCsin 35° = ; .5736 ; BC 57.36100 100CD CDcos 35° = ; .8192 ; CD 81.92100 100

≈ ≈

≈ ≈

The perimeter of the rectangle = 2(57.36) + 2(81.92) = 278.56 cm. Perimeter = 278.56 cm 3. Which of the following trigonometric ratios: sine, cosine, or tangent of an angle can have a

value greater than 1? Why is it that the values of the other two trigonometric ratios can never be greater than 1? Explain. Solution: The tangent of an angle can be greater than 1. The sine of an acute angle of a

right triangle is defined as length of leg opposite the anglelength of hypotenuse

and the cosine of an acute

angle of a right triangle is defined as length of leg adjacent to the anglelength of hypotenuse

. The length of a

leg of a right triangle will always be less than the length of the hypotenuse. If the numerator of a fraction is less than the denominator, the fraction is always less than 1. Therefore, the sine and cosine of an angle will never be greater than 1 by definition of the sine and cosine ratios.

B C

A D

57.36 cm

81.92 cm 100 cm

35°

Trigonometry - D



4. John, an employee of the U.S. Forestry Service has been asked to determine the height of a

tall tree in Wayne National Forest. He uses an angle measuring device to determine the angle of elevation (angle formed by the line of sight to the top of the tree and a horizontal) to be about 33°. He walks off 40 paces to the base of the tree. If each pace is .6 meters, how tall is the tree to the nearest meter? Support your answer by showing your work and including a diagram.

Solution:

≈

≈

htan 33° =24

h.649424

h 16 m

5. Determine the perimeter to the nearest centimeter and the area to the nearest square

centimeter of the triangle shown below. Support your answer by showing your work and giving an explanation.

Sample Solution: Triangle ABC is an isosceles right triangle. The legs have equal lengths, therefore the acute angles each have a measure of 45°. The ratio of the sides of a 45°- 45°- 90° triangle is

1:1: 2 . The length of each leg is .10 10 2= = 5 222

The perimeter of the triangle is .≈10 + 5 2 + 5 2 = 10 +10 2 24 cm

The area of the triangle is

≈ 21 1• 5 2 • 5 2 = • 25 • 2 25 cm .2 2

Sample Solution:

Triangle ABC is an isosceles right triangle. The legs have equal lengths, therefore the acute angles each have a measure of 45°. The lengths of the legs can be found by using the sine and cosine ratios.

∠

≈ ≈

∠

≈ ≈

ACsin B = sin 45° =10

AC = sin 45° •10 .707 •10 7.07 cmABcos B = cos 45° =10

AB = cos 45° •10 .707 •10 7.07 cm

The perimeter of the triangle is 7.07 + 7.07 + 10 = 24.14 ≈ 24 cm. The area of the triangle is .5 • 7.07 • 7.07 = 24.99 ≈ 25 2cm .

10 cm

A C

B

45°

45°

10 cm

A C

B

45°

45°

33° 24 m

h

Trigonometry - D



6. Use what you know about the side lengths of special right triangles to complete the following

table. Express your answers in simplified radical form.

30° 45° 60°

Sin 12

1 2=22

32

Cos 32

1 2=22

12

Tan 1 3=33

1 = 11

3 = 31

45°

45°

60°

30°

Trigonometry - D



Pasta Waves

Name What graphs are created from the distances of the x-axis and y-axis to points on the unit circle? Instructions: 1. Using the unit circle on the “Unit Circle” page, break a piece of pasta so that it is equal to the

vertical distance from the x-axis to the point at 30°. 2. Glue the pasta on the “S(x) Graph” page at the 30°. 3. Repeat this process for each angle on the unit circle. 4. With your pencil, draw a smooth curve that is formed by the ends of the pasta. This is the

graph of S(x). 5. Using the unit circle, break a piece of pasta so that it is equal to the horizontal distance from

the y-axis to the point at 0°. 6. Glue the pasta on the “C(x) Graph” page at 0°. 7. Repeat this process for each angle from the unit circle. 8. With your pencil, draw a smooth curve that is formed by the ends of the pasta. This is the

graph of C(x).

Trigonometry - E



Unit Circle

0º, 360º

30º

45º

60º

90º 120º

135º

150º

180º

210º

225º

240º 270º

300º

315º

330º

Trigonometry - E



S(x) Graph Trigonometry - E

180º 360º

90º 270º

315º 225º

45º 135º

30º 60º

120º 150º

210º 240º

300º 330º

1

-1

0.5

-0.5



C(x) Graph Trigonometry - E

180º 360º

90º 270º

315º 225º

45º 135º

30º 60º

120º 150º

210º 240º

300º 330º

1

-1

0.5

-0.5



Observations and Predictions

1. In the space provided below, list as many similarities as you can between the graphs of S(x) and C(x) that you created.

2. In the space provided below, list as many differences as you can between the graphs

of S(x) and C(x) that you created.

Trigonometry - E



3. Predict what would happen if you continued to measure around the unit circle another

revolution and then pasted your results to the graphs of S(x) and C(x). Draw a sketch of your prediction.

4. Predict what would happen if you continued to measure around the unit circle in a clockwise

fashion and then pasted your results to the graphs of S(x) and C(x). Draw a sketch of your prediction.

Trigonometry - E



t (minutes) x-coordinate y-coordinate 0 100 0

2π

π 32π

2π 52π

3π

Getting in Shape

Name You decide to try jogging to shape up. You are fortunate to have a large neighborhood park nearby that has a circular track with a radius of 100 meters. 1. If you run one lap around the track, how may meters have you traveled? Explain how you determined your answer. 2. You start off averaging a relatively slow rate of approximately 100 meters per minute. How long does it take you to complete one lap? Explain how you determined your answer. You want to improve your speed. Gathering data describing our position on the track as a function of time may be useful. You sketch the track on a coordinate system with the center at the origin. Assume that the starting line has coordinates (100, 0) and that you run counterclockwise. 3. You begin to gather data about your position at various times. Using the preceding diagram and the results from number 2, complete the following table to locate your coordinates at selected times along your path.

(100, 0)

Trigonometry - E



-1.5708 1.5708 3.14159 4.71239 6.28319

4. What patterns do you notice about the numerical data in number 3? Predict your coordinates for the next two time intervals. 5. To better analyze your position at times other than those listed in the previous table. You

decide to make some educated guesses. Consider t = 4π minutes. Use the graph to approximate

the coordinates of your position P when t = 4π minutes, and label these coordinates on the graph.

6. Compute the angle θ given in the graph of number 5 and explain how you arrived at your answer. 7. Gather more information about your position by completing the following table.

8. On the following grid, use the data from number 3 and 7 to plot (t, y). Connect your data pairs to make a smooth graph.

t (minutes) x-coordinate y-coordinate

4π

34π

54π

74π

94π

(100, 0)

Trigonometry - E



9. Use your graphing calculator to plot y = 100 sin x. Make sure your calculator is in radian mode. How does it compare with the graph in number 8? 10. What are the maximum and minimum values of the sine function in problem 9?

The amplitude of a periodic function is defined by: Amplitude = ( )12

M m− , where M is the

maximum output value and m is the minimum output value. 11. What is the amplitude of the sine function in problem 9? How does this relate to equation? 12. Determine the amplitude of each of the following: a. y = 1.5 sin x b. y = 15 sin (x + π) c. y = 3 sin (2x) d. y = -2 sin x 13. Is amplitude ever negative? Why or Why not? 14. Does this same idea of amplitude apply to other trigonometric functions? Which ones? Why or Why not? 15. Write a general statement about the amplitude of the trigonometric functions.

Trigonometry - E



Getting in Shape Answer Key

You decide to try jogging to shape up. You are fortunate to have a large neighborhood park nearby that has a circular track with a radius of 100 meters. 1. If you run one lap around the track, how may meters have you traveled? Explain how you determined your answer. 1 lap = the circumference of the circle C = 2πr = 2π(100 m) = 628 m 2. You start off averaging a relatively slow rate of approximately 100 meters per minute. How long does it take you to complete one lap? Explain how you determined your answer.

t = 628100

= 6.28 min.

You want to improve your speed. Gathering data describing our position on the track as a function of time may be useful. You sketch the track on a coordinate system with the center at the origin. Assume that the starting line has coordinates (100, 0) and that you run counterclockwise. 3. You begin to gather data about your position at various times. Using the preceding diagram and the results from number 2, complete the following table to locate your coordinates at selected times along your path.

t (minutes) x-coordinate y-coordinate0 100 0

2π

0

100

π -100 0 32π

0

-100

2π 100 0 52π

0

100

3π -100 0

(100, 0)

Trigonometry - E



-1.5708 1.5708 3.14159 4.71239 6.28319

4. What patterns do you notice about the numerical data in number 3? Predict your coordinates for the next two time intervals.

The data are cyclic. t = 72π = (0, -100) and t = 4π = (100, 0)

5. To better analyze your position at times other than those listed in the previous table. You

decide to make some educated guesses. Consider t = 4π minutes. Use the graph to approximate

the coordinates of your position P when t = 4π minutes, and label these coordinates on the graph.

t = 4π ; d = 100(

4π ) = 25π ; =25 1

200 8ππ

lap

6. Compute the angle θ given in the graph of number 5 and explain how you arrived at your answer.

θ = 4π = 45o 1

8(2π) =

4π 1

8(360) = 45o

7. Gather more information about your position by completing the following table.

8. On the following grid, use the data from number 3 and 7 to plot (t, y). Connect your data pairs to make a smooth graph.

t (minutes) x-coordinate y-coordinate

4π 70.71 70.71

34π -70.71 70.71

54π -70.71 -70.71

74π 70.71 -70.71

94π 70.71 70.71

(100, 0)

(70.7, 70.7) (-70.7, 70.7)

(-70.7, -70.7) (70.7, -70.7)

Trigonometry - E



9. Use your graphing calculator to plot y = 100 sin x. Make sure your calculator is in radian mode. How does it compare with the graph in number 8? The graphs are the same. 10. What are the maximum and minimum values of the sine function in problem 9? 100 is the maximum value and -100 is the minimum value

The amplitude of a periodic function is defined by: Amplitude = ( )12

M m− , where M is the

maximum output value and m is the minimum output value. 11. What is the amplitude of the sine function in problem 9? How does this relate to equation? Amplitude = 100 It is the coefficient in the equation. 12. Determine the amplitude of each of the following: a. y = 1.5 sin x b. y = 15 sin (x + π) c. y = 3 sin (2x) d. y = -2 sin x 1.5 15 3 2 13. Is amplitude ever negative? Why or Why not? Amplitude is never negative. Amplitude is one half of the maximum minus the minimum. 14. Does this same idea of amplitude apply to other trigonometric functions? Which ones? Why or Why not? Amplitude also applies to the cosine function. The other functions do not have an absolute maximum and minimum. 15. Write a general statement about the amplitude of the trigonometric functions. Amplitude = a M m= −

Trigonometry - E



-1.5708 1.5708 3.14159 4.71239 6.28319

Speeding Up

Name After a lot of practice, you finally speed up on your circular track of radius 100 meters. You achieve your personal goal of 200 meters per minute. 1. Running at 200 meters per minute counterclockwise, how long does it take you to complete

one lap? 12

of a lap? 14

of a lap?

2. Use the results of number 1 to complete the following table that indicates coordinates at selected special points along your path. t (minutes) x-coordinate y-coordinate

0 100 0

4π

2π

34π

π 54π

32π

74π

2π 3. On the following grid, use the data from number 2 to plot (t, y). Connect your data pairs to make a smooth graph. 4. What is the amount of time it takes for the graph of problem 3 to complete one full cycle?

Trigonometry - E



-1.5708 1.5708 3.14159 4.71239 6.28319

5. When you doubled your speed, what effect did that have on the amount of time to complete one cycle? Explain how you determined your answer. 6. Use your graphing calculator to plot y = 100 sin 2x. How does this compare to the graph in number 3? 7. The period of a trigonometric function is the shortest time (distance) to complete one full cycle. Determine the period of each of the following: a. y = 100 sin x b. y = 100 sin (2x) c. y = 100 sin (3x) 8. On the following grid, use the data from number 2 to plot (t, x). Connect your data pairs to make a smooth graph. 9. What equation should you put in the calculator to produce the graph from number 8? 10. If you tripled your speed on the track from the original 100 meters per minute, what effect would this have on the amount of time to compete one lap? What equations would describe the x and y coordinates of your position? 11. Determine the period of each of the following functions: a. y = 100 cos x b. y = 100 cos (2x) c. y = 100 cos (3x)

Trigonometry - E



-1.5708 1.5708 3.14159 4.71239 6.28319

Speeding Up Answer Key

After a lot of practice, you finally speed up on your circular track of radius 100 meters. You achieve your personal goal of 200 meters per minute. 1. Running at 200 meters per minute counterclockwise, how long does it take you to complete

one lap? 12

of a lap? 14

of a lap?

One lap = 200π meters; t = 200200

π = π min; 12

lap ; t = 2π min; 1

4 lap; t =

4π min

2. Use the results of number 1 to complete the following table that indicates coordinates at selected special points along your path. t (minutes) x-coordinate y-coordinate

0 100 0

4π 0 100

2π -100 0

34π 0 -100

π 100 0 54π 0 100

32π -100 0

74π 0 -100

2π 100 0 3. On the following grid, use the data from number 2 to plot (t, y). Connect your data pairs to make a smooth graph. 4. What is the amount of time it takes for the graph of problem 3 to complete one full cycle? π minutes

Trigonometry - E



-1.5708 1.5708 3.14159 4.71239 6.28319

5. When you doubled your speed, what effect did that have on the amount of time to complete one cycle? Explain how you determined your answer. Doubling the speed cuts the time in half. 6. Use your graphing calculator to plot y = 100 sin 2x. How does this compare to the graph in number 3? The graphs are the same. 7. The period of a trigonometric function is the shortest time (distance) to complete one full cycle. Determine the period of each of the following: a. y = 100 sin x b. y = 100 sin (2x) c. y = 100 sin (3x)

2π π 23π

8. On the following grid, use the data from number 2 to plot (t, x). Connect your data pairs to make a smooth graph. 9. What equation should you put in the calculator to produce the graph from number 8? y = 100 cos (2x) 10. If you tripled your speed on the track from the original 100 meters per minute, what effect would this have on the amount of time to compete one lap? What equations would describe the x and y coordinates of your position?

The time would be reduced by a factor of 3 from 2π min to 23π min.

x = 100 cos (3t) and y = 100 sin (3t) 11. Determine the period of each of the following functions: a. y = 100 cos x b. y = 100 cos (2x) c. y = 100 cos (3x)

2π π 23π

Trigonometry - E



3.14159 6.28319 9.42478

Running With a Friend

Name While jogging, you become good friends with another runner, who started out like you, with a speed of 100 meters per minute on the 100 meter track. You enjoy running together but prefer to keep a healthy distance between each other during the run. You start at (100, 0) and when you arrive at (0, 100), your friend starts at (100,0). 1. Assume that you both maintain the same speed of 100 meters per minute. How far ahead of your friend are you? How long after you start does your friend wait before starting. 2. Complete the following table to give coordinates for both you and your friend at selected points along the track.

3. On the following grid, use the data from number 2 to plot (t, y) for both you and your friend. Connect the points to make a smooth curve. 4. What is the relationship between the two graphs? The displacement, or phase shift, of a graph is the smallest movement (left or right) necessary for the first graph to match the second graph.

t (minutes) Your x-coordinate

Your y-coordinate

Friend x-coordinate

Friend y-coordinate

0 100 0 ---- ----

2π

100

0

π 32π

2π 52π

Trigonometry - E



5. Which graph is the displaced graph in number 3? What is the displacement? 6. Would you expect the same type of relationship between the two graphs representing the x-coordinates? Why or Why Not? Displacement is defined for the cosine function in the same manner. 7. If your x-coordinate is given by 100 cos t, predict the defining equation of your friend’s x-coordinate. In general, in the following functions defined by y = a sin(bx + c) and y = a cos(bx + c), the values for b and c affect the displacement, or phase shift, of the function. The phase shift is

given by cb

− .

8. If cb

− is negative, then the shift is to the __________________. When cb

− is positive, then

the shift is to the __________________. 9. In this activity, your friend’s y-coordinate is given by y = a sin(bx + c), where a = 100 and b = 1. Calculate c. 10. For the following functions, identify the amplitude, period, and displacment:

a. y = 0.7 cos(2x + 2π ) b. y = 3 sin(x – 1) c. y = - 2.5 sin(0.4x +

3π )

Trigonometry - E



-1.5708 1.5708 3.14159 4.71239 6.28319 7.85398 9.42478

Running With a Friend Answer Key

While jogging, you become good friends with another runner, who started out like you, with a speed of 100 meters per minute on the 100 meter track. You enjoy running together but prefer to keep a healthy distance between each other during the run. You start at (100, 0) and when you arrive at (0, 100), your friend starts at (100,0). 1. Assume that you both maintain the same speed of 100 meters per minute. How far ahead of your friend are you? How long after you start does your friend wait before starting.

You are 14

lap ahead. m = 50π m = 157 min

2. Complete the following table to give coordinates for both you and your friend at selected points along the track. t (minutes) Your

x-coordinate Your

y-coordinateFriend

x-coordinateFriend

y-coordinate0 100 0 ---- ----

2π

0

100

100

0

π -100 0 0 100 32π

0

-100

-100

0

2π 100 0 0 -100 52π

0

100

100

0

3. On the following grid, use the data from number 2 to plot (t, y) for both you and your friend. Connect the points to make a smooth curve. 4. What is the relationship between the two graphs?

The graph for the friend is shifted 2π units to the right.

The displacement, or phase shift, of a graph is the smallest movement (left or right) necessary for the first graph to match the second graph.

Trigonometry - E



5. Which graph is the displaced graph in number 3? What is the displacement?

The friend’s graph is displaced by 2π minutes.

6. Would you expect the same type of relationship between the two graphs representing the x-coordinates? Why or Why Not? Yes. By looking at the table, you can see that the x-coordinates are the same for values of t

that differ by 2π minutes.

Displacement is defined for the cosine function in the same manner. 7. If your x-coordinate is given by 100 cos t, predict the defining equation of your friend’s x-coordinate.

x = 100 cos(t - 2π )

In general, in the following functions defined by y = a sin(bx + c) and y = a cos(bx + c), the values for b and c affect the displacement, or phase shift, of the function. The phase shift is

given by cb

− .

8. If cb

− is negative, then the shift is to the __left______________. When cb

− is positive,

then the shift is to the ___right___________. 9. In this activity, your friend’s y-coordinate is given by y = a sin(bx + c), where a = 100 and b = 1. Calculate c.

c = - 2π

10. For the following functions, identify the amplitude, period, and displacment:

a. y = 0.7 cos(2x + 2π ) b. y = 3 sin(x – 1) c. y = - 2.5 sin(0.4x +

3π )

Amplitude = 0.7 Amplitude = 3 Amplitude = 2.5 Period = π Period = 2π Period = 5π

Displacement = 4π− Displacement = 1 Displacement = 5

6π−

Trigonometry - E



Sine Curve Equation

Name Match the sine curve equations on the right to their characteristics on the left. Place the letter of the matching equation on the blank before its characteristics. Amplitude Period Displacement Sine Curve Equations

__________ 1. 1 2π 4π units left a. sin .4(x +

4π )

__________ 2. 2 π 2π units right b. 1.5sin 4(x -

6π )

__________ 3. 1.5 2π

6π units right c. 2sin 2(x -

6π )

__________ 4. 1.5 2π 6π units left d. sin 3(x + π)

__________ 5. 1 23π π units left e. sin (x +

4π )

__________ 6. 2 4π 3π units right f. 1.5sin .2(x – π)

__________ 7. 1 34π

2π units right g. sin 2

3(x +

2π )

__________ 8. 1 5π 4π units left h. 1.5sin 2(x – π)

__________ 9. 1.5 π π units right i. 1.5sin (x + 6π )

__________ 10. 2 2π 2

3π units left j. 2sin 2(x -

2π )

__________ 11. 1 3π 2π units left k. sin 8

3(x -

2π )

__________ 12. 2 π 6π units right l. sin 3(x -

6π )

__________ 13. 1.5 4π 2π units left m. 2sin 4(x + 23π )

__________ 14. 1.5 10π π units right n. 2sin 2(x - 4π )

o. 2sin 12

(x - 3π )

p. 1.5sin (x - 6π )

q. 1.5sin 12

(x + 2π)

Trigonometry - E



Sine Curve Equation Answer Key

Match the sine curve equations on the right to their characteristics on the left. Place the letter of the matching equation on the blank before its characteristics. Amplitude Period Displacement Sine Curve Equations

____E____ 1. 1 2π 4π units left a. sin .4(x +

4π )

____J____ 2. 2 π 2π units right b. 1.5sin 4(x -

6π )

____B____ 3. 1.5 2π

6π units right c. 2sin 2(x -

6π )

____I____ 4. 1.5 2π 6π units left d. sin 3(x + π)

____D____ 5. 1 23π π units left e. sin (x +

4π )

____O____ 6. 2 4π 3π units right f. 1.5sin .2(x – π)

____K____ 7. 1 34π

2π units right g. sin 2

3(x +

2π )

____A____ 8. 1 5π 4π units left h. 1.5sin 2(x – π)

____H____ 9. 1.5 π π units right i. 1.5sin (x + 6π )

____M____ 10. 2 2π 2

3π units left j. 2sin 2(x -

2π )

____G____ 11. 1 3π 2π units left k. sin 8

3(x -

2π )

____C____ 12. 2 π 6π units right l. sin 3(x -

6π )

____Q____ 13. 1.5 4π 2π units left m. 2sin 4(x + 23π )

____F____ 14. 1.5 10π π units right n. 2sin 2(x - 4π )

o. 2sin 12

(x - 3π )

p. 1.5sin (x - 6π )

q. 1.5sin 12

(x + 2π)

Trigonometry - E



Finding the Inverse Without a Calculator

Name Use the unit circle to find the value of each of the following in both radians and degrees.

1. sin-1 22

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

2. cos-1 22

3. cot-1 1 4. csc-1 1 5. arcsec 2 6. tan-1 (- 3 )

7. arcsin 32

8. csc-1 2

9. arccos 22

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

10. arctan 33

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

Trigonometry - F



Finding the Inverse Without a Calculator Answer Key

Use the unit circle to find the value of each of the following in both radians and degrees on the interval [0, 2π].

1. sin-1 22

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

2. cos-1 22

315o, 225o 45o, 315o 7 5,4 4π π 7,

4 4π π

3. cot-1 1 4. csc-1 1 45o, 225o 90o

5,4 4π π

2π

5. arcsec 2 6. tan-1 (- 3 ) 60o, 300o 330o, 150o

5,3 3π π 11 5,

6 6π π

7. arcsin 32

8. csc-1 2

60o, 120o 30o, 150o

2,3 3π π 5,

6 6π π

9. arccos 22

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

10. arctan 33

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

135o, 225o 330o, 150o 3 5,4 4π π 11 5,

6 6π π

Trigonometry - F



-6.28319 -3.14159 3.14159 6.28319

-4.7124

-3.1416

-1.5708

1.5708

3.1416

4.7124

F

-6.28319 -3.14159 3.14159 6.28319

-4.7124

-3.1416

-1.5708

1.5708

3.1416

4.7124

E

-3.1416 3.1416

-4.7124

-3.1416

-1.5708

1.5708

3.1416

4.7124

D

-6.28319 -3.14159 3.14159 6.28319

-4.7124

-3.1416

-1.5708

1.5708

3.1416

4.7124

C

-6.28319 -3.14159 3.14159 6.28319

-4.7124

-3.1416

-1.5708

1.5708

3.1416

4.7124

A

-6.28319 -3.14159 3.14159 6.28319

-4.7124

-3.1416

-1.5708

1.5708

3.1416

4.7124

B

Match the Inverse

Name Match the graph of the right with the equation on the left. _____ 1. y = sin-1x _____ 2. y = cos-1x _____ 3. y = tan-1x _____ 4. y = csc-1x _____ 5. y = sec-1x _____ 6. y = cot-1x

Trigonometry - F



-6.28319 -3.14159 3.14159 6.28319

-4.7124

-3.1416

-1.5708

1.5708

3.1416

4.7124

F

-6.28319 -3.14159 3.14159 6.28319

-4.7124

-3.1416

-1.5708

1.5708

3.1416

4.7124

E

-3.1416 3.1416

-4.7124

-3.1416

-1.5708

1.5708

3.1416

4.7124

D

-6.28319 -3.14159 3.14159 6.28319

-4.7124

-3.1416

-1.5708

1.5708

3.1416

4.7124

C

-6.28319 -3.14159 3.14159 6.28319

-4.7124

-3.1416

-1.5708

1.5708

3.1416

4.7124

A

-6.28319 -3.14159 3.14159 6.28319

-4.7124

-3.1416

-1.5708

1.5708

3.1416

4.7124

B

Match the Inverse Answer Key

Match the graph of the right with the equation on the left. ___E__ 1. y = sin-1x ___A__ 2. y = cos-1x ___F__ 3. y = tan-1x ___D__ 4. y = csc-1x ___C__ 5. y = sec-1x ___B__ 6. y = cot-1x

Trigonometry - F



Daylight Hours

Name The Astronomical Applications Department of the U. S. Naval Observatory gives the time of sunrise and sunset for each day in the year. It is available at http://aa.usno.navy.mil/data/docs/RS_OneYear.html. Your group will be assigned a city by your teacher. Each group will make a scatterplot of the day of the year vs. length of day for the assigned city. You should enter the day of the year into L1, the time of sunrise into L2, the time of sunset into L3, and L3-L2 into L4, and then graphing L1 vs. L4. Choose every tenth day. Be careful to choose each tenth day. Do not use the days of the month. The entries in L1 should be 10, 20, 30, 40, …, 360. If you wish to divide the work up among your group, all group members should follow the above instructions for their portion of the calendar. Then each one should transfer L1 into L5 and L4 into L6. Then, one at a time, transfer L5 and L6 onto one of the calculators. After each transfer, use Augment to add all of the days to one list and all of the daylight hours into one list. Augment is accessed by LIST, OPS, 9: augment( The commands are 1. Model the statplot using a sine function. 2. Estimate the amplitude of the function. On what days of the year, should the maximum and minimum occur? (You can use the points from your graph or the exact days to find the amplitude.) 3. Estimate the phase shift of the function by estimating the point of inflection of the graph. On what days of the year should the points of inflection occur? (You can use the points from your graph or the exact days to find the phase shift.) 4. What is the vertical shift of your graph? What does this represent in terms of hours of daylight? 5. Write the equation for your model. Enter it into Y1 and check it for accuracy. Sketch your scatterplot and the graph of your model. 6. Compare your graph to those of the other two cities. Why are they different?

Trigonometry - G



Daylight Hours Answer Key

The Astronomical Applications Department of the U. S. Naval Observatory gives the time of sunrise and sunset for each day in the year. It is available at http://aa.usno.navy.mil/data/docs/RS_OneYear.html. Your group will be assigned a city by your teacher. Each group will make a scatterplot of the day of the year vs. length of day for the assigned city. You should enter the day of the year into L1, the time of sunrise into L2, the time of sunset into L3, and L3-L2 into L4, and then graphing L1 vs. L4. Choose every tenth day. Be careful to choose each tenth day. Do not use the days of the month. The entries in L1 should be 10, 20, 30, 40, …, 360. If you wish to divide the work up among your group, all group members should follow the above instructions for their portion of the calendar. Then each one should transfer L1 into L5 and L4 into L6. Then, one at a time, transfer L5 and L6 onto one of the calculators. After each transfer, use Augment to add all of the days to one list and all of the daylight hours into one list. Augment is accessed by LIST, OPS, 9: augment( The commands are 1. Model the statplot using a sine function. ANSWERS WILL VARY 2. Estimate the amplitude of the function. On what days of the year, should the maximum and minimum occur? (You can use the points from your graph or the exact days to find the amplitude.) The amplitudes vary according to the latitude of the city, the farther from the equator, the larger the amplitude. The max should occur on the summer solstice, June 21. The min should occur on the winter solstice, December 21 or 22, depending on the year. 3. Estimate the phase shift of the function by estimating the point of inflection of the graph. On what days of the year should the points of inflection occur? (You can use the points from your graph or the exact days to find the phase shift.) The points of inflection should occur on the vernal equinox, March 21 or 22 and the autumnal equinox, September 22 or 23. 4. What is the vertical shift of your graph? What does this represent in terms of hours of daylight? This is number of hours of daylight at the equinox. It should be the same on all the graphs. 5. Write the equation for your model. Enter it into Y1 and check it for accuracy. Sketch your scatterplot and the graph of your model. Answers will vary. 6. Compare your graph to those of the other two cities. Why are they different? Different latitudes.

Trigonometry - G



Tic-Tac Trig - 4 in a Row

Name

Directions for Play:

Objective: The object of the game is similar to that of tic-tac-toe; the winner is the first of two players to place four tokens in a row, either vertically, horizontally, or diagonally.

Materials: The materials necessary to play Trig tic-tac-times include a factor board and a game board (see the next page) and forty translucent tokens of two different colors. One token of each color is used as the factor marker, and the remainder are used as game tokens. The game board should be laminated so that it can be saved from year to year. The tokens should be stored in a bag so that they don't get lost.

Method of play: Player 1 begins the game by placing a factor marker and one of player 2's factor markers on any factors on the factor board. The product of these factors determines the placement of player 1's game token. For example, player 1 could place a factor marker on sin(-x) and player 2's marker on cot(x). Player 1 then would place a game token on -cos(x) because [sin(-x)][cot(x)] = -cos(x). Note: factor markers can be placed on the same factor, resulting in squared factors.

Player 2 can move only player 2's factor marker (player l's marker remains in place) to another factor on the factor board. In this example, player 2 could move her factor marker to csc(x). The product of these new factors determines the placement of player 2's game token. In this example, player 2 would place a game token on the product of sin(-x) (player l's marker) and csc(x) (player 2's marker), or -1, on the game board.

Players must use a strategy of working backward to determine which products combined with the available factors will win the game. These same problem-solving strategies become a part of the defensive play of the game when a player wishes to block an opponent.

Trigonometry - H



Tic-Tac Trig - 4 in a Row

Game Board A B C D E

- cos x

-1

1

csc x cot x

- cot x csc x

sin x tan x

sin x

- 2sin

cosxx

- sin x tan x

cos x

1 – cos2x

sec2x - 1

12

sin (2x)

tan2x + 1

cos2x – 1

- csc x

sec2 x sin x

1 – sec2x

sec x sin2x

1

csc2 x – 1

cot x

cos x

csc2x – 1

cot2x + 1

1

sec2 x – 1

1 – sin2x

csc x tan x

1 – csc2x

2cos

sinx

x−

- cos x

sec x cot x

-1

- sin x

- tan x

tan x

- 1

cos2x csc x

sin2x

csc x sec x

- sec x tan x

- sin x cos x

1

- sec x

Factor Board

sin (-x)

tan(-x)

sin x

cos x

tan x

cot x

sec x

csc x

cot (-x)

Trigonometry - H



Tic-Tac Trig - 4 in a Row Answer Key

THIS IS FOR TEACHERS ONLY!! Listed below are the possible solutions for each of the 45 squares in the game board. A-1 sin(-x) cot(x) sin(x) cot(-x) B-1 tan(-x) cot(x) tan(x) cot(-x) sin(-x) csc(x) C-1 cos(x) sec(x) tan(-x) cot(-x) tan(x) cot(x) sin(x) csc(x) D-1 cot(x) csc(x) E-1 cot(-x) csc(x) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - A-2 sin(-x) tan(-x) sin(x) tan(x) B-2 cos(x) tan(x) C-2 sin(-x) tan(x) sin(x) tan(-x) D-2 sin(-x) tan(x) sin(x) tan(-x) E-2 sin(-x) cot(-x) sin(x) cot(x) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - A-3 sin(x) sin(x) sin(-x) sin(-x) B-3 tan(-x) tan(-x) tan(x) tan(x)

C-3 12

sin(2x)

D-3 tan2x + 1 E-3 cos2x – 1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - A-4 cot(-x) sec(x) B-4 tan(x) sec(x) C-4 tan(x) tan(-x) D-4 sin(-x) tan(-x) sin(x) tan(x) E-4 cos(x) sec(x) tan(-x) cot(-x) tan(x) cot(x) sin(x) csc(x) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - A-5 cot(x) cot(x) cot(-x) cot(-x) B-5 cos(x) csc(x) C-5 sin(-x) cot(-x) sin(x) cot(x) D-5 cot(x) cot(x) cot(-x) cot(-x) E-5 csc(x) csc(x) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - A-6 cos(x) sec(x) tan(-x) cot(-x) tan(x) cot(x) sin(x) csc(x) B-6 tan(-x) tan(-x) tan(x) tan(x) C-6 cos(x) cos(x) D-6 tan(x) csc(x) E-6 cot(x) cot(-x) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - A-7 cos(x) cot(-x) B-7 sin(-x) cot(x) sin(x) cot(-x) C-7 cot(x) sec(x) D-7 tan(-x) cot(x) tan(x) cot(-x) sin(-x) csc(x) E-7 cos(x) tan(-x)

Trigonometry - H



- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - A-8 sin(-x) sec(x) B-8 sin(x) sec(x) C-8 tan(-x) cot(x) tan(x) cot(-x) sin(-x) csc(x) D-8 cos(x) cot(x) E-8 sin(x) sin(x) sin(-x) sin(-x) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - A-9 sec(x) csc(x) B-9 tan(-x) sec(x) C-9 sin(-x) cos(x) D-9 cos(x) sec(x) tan(-x) cot(-x) tan(x) cot(x) sin(x) csc(x) E-9 tan(-x) csc(x)

Trigonometry - H



Law of Sines Spaghetti Investigation

Name 1. Measure each of these line segments and break pieces of spaghetti equal in length to each. a. e. c. d. 2. Use combinations of three sides from a, e, c, and d to try to make triangles. Which combinations of three of these can you use to make a triangle? Why do the others not work? 3. Use side a as the base and c as the hypotenuse to make a right triangle. Break a piece of spaghetti to make the other leg of the triangle. Name this side b. Find the length. Compare the answer with the length of the piece of spaghetti that you broke, side b. Draw the triangle and label the sides. Use right triangle trigonometry to find the degree measure of the angle. 4. Use the pipe cleaner to form an angle congruent to the angle B. Make one side of the angle the same length as c. Tape a piece of spaghetti to the other side. Place the piece of spaghetti for side b to recreate the right triangle from #3. Label the vertices of the triangle ABC.

Trigonometry - I Trigonometry - I



5. Break a piece of spaghetti slightly longer than b. Place one end of the spaghetti at vertex A. Align the spaghetti so that it touches the base of the triangle. How many ways can you do this? Explain this in terms of the length of the piece of spaghetti and side b. 6. Draw the two possibilities. Use the Law of Sines to solve each of the triangles. 7. Set the last piece of spaghetti aside and break another piece slightly shorter than b. How many triangles can you make now? Explain this in terms of the length of the piece of spaghetti and side b. 8. Set the second piece of spaghetti aside and break a third piece a little longer than side c. How many triangles can you make using this for the side from vertex A? 9. Draw the triangle and use the Law of Sines to solve the triangle.

Trigonometry - I



You have been investigating what is known as The Ambiguous Case of the Law of Sines. The ambiguous case occurs when you are given an acute angle B, the side b, and another side a that is longer than b. To determine the number of solutions:

• Check the value of a sin B

• If b < asinB , there is no triangle.

• If asinB < b < a , there are two solutions.



Law of Sines Spaghetti Investigation Answer Key

1. Measure each of these line segments and break pieces of spaghetti equal in length to each. a. 4.7 cm e. 7 cm c. d. 5.7 cm 9.5 cm 2. Use combinations of three sides from a, e, c, and d to try to make triangles. Which combinations of three of these can you use to make a triangle? Why do the others not work? All of the sides work to make triangles because they all work in the triangle inequality 3. Use side a as the base and c as the hypotenuse to make a right triangle. Break a piece of spaghetti to make the other leg of the triangle. Name this side b. Find the length. Compare the answer with the length of the piece of spaghetti that you broke, side b. Draw the triangle and label the sides. Use right triangle trigonometry to find the degree measure of the angle. Angle B measures 60.348º 4. Use the pipe cleaner to form an angle congruent to the angle B. Make one side of the angle the same length as c. Tape a piece of spaghetti to the other side. Place the piece of spaghetti for side b to recreate the right triangle from #3. Label the vertices of the triangle ABC.

b 9.5 B

4.7

A spaghetti pipe cleaner B C pipe cleaner

Trigonometry - I Trigonometry - I



5. Break a piece of spaghetti slightly longer than b. Place one end of the spaghetti at vertex A. Align the spaghetti so that it touches the base of the triangle. How many ways can you do this? Explain this in terms of the length of the piece of spaghetti and side b. Two ways, one to the left and one to the right. 6. Draw the two possibilities. Use the Law of Sines to solve each of the triangles. ANSWERS WILL VARY 7. Set the last piece of spaghetti aside and break another piece slightly shorter than b. How many triangles can you make now? Explain this in terms of the length of the piece of spaghetti and side b. No triangles. Side b does not reach the other side. 8. Set the second piece of spaghetti aside and break a third piece a little longer than side c. How many triangles can you make using this for the side from vertex A? Just one.

A C1 C2 B

Trigonometry - I



9. Draw the triangle and use the Law of Sines to solve the triangle. Answers will vary. You have been investigating what is known as The Ambiguous Case of the Law of Sines. The ambiguous case occurs when you are given an acute angle B, the side b, and another side a that is longer than b. To determine the number of solutions:

• Check the value of a sin B

• If b < asinB , there is no triangle.

• If asinB < b < a , there are two solutions.

A C B



Law of Sines: Real-life Applications

Name 1. A power company would like to run a high-voltage power line across a canyon. It must extend from point A on the south side of the canyon to point C on the north side. A surveyor has marked point B on the south side of the canyon 100 feet from point A and has determined that the measure of angle CAB is 42º, and the measure of angle ABC is 110º. What is the minimum length of power line that will be needed? 2. A motorist is traveling on a straight and level highway at a constant speed of 60 miles per hour. A mountain top with an angle of elevation of 10º is visible straight ahead. Five minutes later, the angle of elevation to the top of the mountain is 25º. Find the height of the mountain top (in feet) relative to the highway? 3. Two guy wires for a radio tower make angles with the horizontal of 58º and 49º. If the ground anchors for each wire are 150 feet apart, find a. the length of each wire, assuming the wires are taught b. the height of the tower.

Trigonometry - I



4. An airplane is flying at a constant altitude of 3650 feet and a constant speed of 660 feet per second (approximately 450 miles per hour) on a path that will take it directly over the Eiffel Tower. At a certain point, the angle of depression to the top of the tower is 22º. Four seconds later, the angle of depression to the top of the tower is 34º. Estimate the height of the Eiffel Tower. 5. Two Coast Guard stations are located 10 miles apart on a coastline that runs north-south. A distress signal is received from a ship with a bearing of N 43º E from the southern station and a bearing of S 56º E from the northern station. How far is the ship from each station? How far is the ship from shore?

Trigonometry - I



Law of Sines: Real-life Applications Answer Key

1. A power company would like to run a high-voltage power line across a canyon. It must extend from point A on the south side of the canyon to point C on the north side. A surveyor has marked point B on the south side of the canyon 100 feet from point A and has determined that the measure of ∠ CAB is 42º, and the measure of ∠ ABC is 110º. What is the minimum length of power line that will be needed?

sin 28 sin110100 AC

=

C A AC = 200.16 ft 100 ft B 2. A motorist is traveling on a straight and level highway at a constant speed of 60 miles per hour. A mountain top with an angle of elevation of 10º is visible straight ahead. Five minutes later, the angle of elevation to the top of the mountain is 25º. Find the height of the mountain top (in feet) relative to the highway? M 10o 155o 25o A 26400 ft B H

sin15 sin1026400 BM

= BM = 17712.42 ft sin 2517712.42

MH= MH = 7485.59 ft

3. Two guy wires for a radio tower make angles with the horizontal of 58º and 49º. If the ground anchors for each wire are 150 feet apart, find a. the length of each wire, assuming the wires are taught and b. the height of the tower. T 73o sin 73 sin 58 sin 73 sin 49150 150TB TA

= = h

58o z 49o TB = 133.02 ft TA = 118.38 ft A 150 ft B z2 + h2 = 118.382 and (150 – z)2 + h2 = 133.022 Use these two equations to solve for z and h. z = 62.73 ft and h = 100.39 ft (h is the height of the tower)

Trigonometry - I



4. An airplane is flying at a constant altitude of 3650 feet and a constant speed of 660 feet per second (approximately 450 miles per hour) on a path that will take it directly over the Eiffel Tower. At a certain point, the angle of depression to the top of the tower is 22º. Four seconds later, the angle of depression to the top of the tower is 34º. Estimate the height of the Eiffel Tower. 2640 ft

A B C T

5. Two Coast Guard stations are located 10 miles apart on a coastline that runs north-south. A distress signal is received from a ship with a bearing of N 43º E from the southern station and a bearing of S 56º E from the northern station. How far is the ship from each station? How far is the ship from shore? N 56o L H 10 mi 43o S

sin 81 sin 43 sin 81 sin 5610 10SH NH

= =

NH = 6.90 mi SH = 8.39 mi

sin 438.39LH=

LH = 5.72 mi

∠ BAT = 22o; ∠ ABT = 146o; ∠ BTA = 12o sin12 sin 222640 TB

= TB = 4756.64 ft

sin 344756.64

TC= TC = 2659.88 ft

Height of Tower = 3650 – 2659.88 = 990.12 ft

Trigonometry - I



Sines Just Aren’t Enough

Name Note: Round all final answers to the nearest hundredth. 1. Robbie the Robot is placed on a level practice field facing due north. He moves 50 feet in that direction. He then stops and rotates clockwise through 148º (so that he is facing approximately southeast. Robbie then moves 65 feet in that new direction and stops. a. How far is Robbie from his starting point? b. How many degrees clockwise should Robbie rotate in order to be facing the starting point? 2. Two ships start from the same point and sail in different directions, one on a course of 40º clockwise from north at a rate of 6 miles per hour and the other on a course of 150º clockwise from north at a rate of 8 miles per hour. How far apart are the ships after 2.5 hours?

Trigonometry - I



3. In a major-league baseball diamond, the bases form a square with side length 90 feet. The pitcher’s mound is 60.5 feet from home plate. Find the distance from the pitcher’s mound to each of the other three bases. 4. A 10-foot by 12-foot by 8-foot rectangular room is to be partitioned using a triangular piece of canvas. Each of the vertices of the triangle is to be anchored at three corners of the room, as shown in the figure below. Find the measure of each vertex angle of the triangle. 10 ft B 8 ft C A 12 ft 5. Margaret and Elizabeth live 1 mile apart. While talking to each other on the phone during an electrical storm, they both notice a bolt of lightning in the sky between their two homes. Margaret hears the thunder 5 seconds after the flash, while Elizabeth hears the thunder 4 seconds after the flash. If the speed of sound is 1088 feet per second, describe the location of the lightning bolt relative to Margaret’s position.

Trigonometry - I



Sines Just Aren’t Enough Answer Key

Note: Round all final answers to the nearest hundredth. 1. Robbie the Robot is placed on a level practice field facing due north. He moves 50 feet in that direction. He then stops and rotates clockwise through 148º (so that he is facing approximately southeast. Robbie then moves 65 feet in that new direction and stops. a. How far is Robbie from his starting point? 32o a2 = 502 + 652 – 2(50)(65)(cos 32) a2 = 1212.687375 65 ft 50 ft a = 34.82 ft b. How many degrees clockwise should Robbie rotate in order to be facing the starting point?

65 57.28sin sin 32B

=

65(sin 32) = 57.28(sin B) 65(sin 32)sin

57.28B =

B = 36.97o 2. Two ships start from the same point and sail in different directions, one on a course of 40º clockwise from north at a rate of 6 miles per hour and the other on a course of 150º clockwise from north at a rate of 8 miles per hour. How far apart are the ships after 2.5 hours? a2 = 152 + 202 – 2(15)(20)(cos 110) 15 mi a2 = 830.2121 a = 28.81 mi 110o 20 mi

Trigonometry - I



3. In a major-league baseball diamond, the bases form a square with side length 90 feet. The pitcher’s mound is 60.5 feet from home plate. Find the distance from the pitcher’s mound to each of the other three bases. a2 = 902 + 60.52 – 2(90)(60.5)(cos 45) 90 ft a2 = 4059.857153 b a = 63.72 ft. (pitcher’s mound to 1st or 3rd base) a __ 90 ft 60.5 ft 902 + 902 = c2 16200 = c2 127.28 ft = c b = 127.28 ft – 60.5 ft = 66.78 ft (pitcher’s mound to 2nd base) 4. A 10-foot by 12-foot by 8-foot rectangular room is to be partitioned using a triangular piece of canvas. Each of the vertices of the triangle is to be anchored at three corners of the room, as shown in the figure below. Find the measure of each vertex angle of the triangle. 10 ft B BC = 8 ft AC = 2 212 10+ = 244 2 61=

8 ft C AB = ( )228 244 308 2 77+ = =

A 12 ft ∠ BCA = 90o

tan A = 8244

= .5121475197

∠ BAC = 27.12o ∠ CBA = 90 – 27.12 = 62.88o 5. Margaret and Elizabeth live 1 mile apart. While talking to each other on the phone during an electrical storm, they both notice a bolt of lightning in the sky between their two homes. Margaret hears the thunder 5 seconds after the flash, while Elizabeth hears the thunder 4 seconds after the flash. If the speed of sound is 1088 feet per second, describe the location of the lightning bolt relative to Margaret’s position. L 43522 = 54402 + 52802 – 2(5440)(5280)(cos M) cos M = .6707486631 M = 47.875o from the line of sight 5440 ft 4352 ft M E 5280 ft

Trigonometry - I



A Variety of Problems

Name Solve each of the following problems using trigonometric properties. 1. Joshua is flying a kite with 135 feet of string out. The angle of elevation of the kite varies from 45 degrees to 60 degrees during the flying time. What is the range of heights the kite will fly during this time? 2. A jet plane is descending from 6000 feet above ground into DFW airport. The angle of depression between the pilot’s line of sight and the control tower is 5o. How many miles is the plane from the control tower? 3. One of the tallest mountains in Alaska is 15,308 feet above sea level, just 14.25 miles from the Pacific coast. What is the angle of elevation from the summit of the mountain to the shoreline of the Pacific Coast?

Trigonometry - J



4. What is the measure of the angle of depression from a tower that is 120 feet tall to a point that is 525 feet away from the tower on level ground? 5. A golfer is standing at the tee, looking up to the green on a hill. If the tee is 36 yards lower than the green and the angle of elevation from the tee to the hole is 12o. Find the distance from the tee to the hole. 6. From the top of a roller coaster, 60 yards above the ground, a rider looks down and sees the merry-go-round and Ferris wheel. If the angles of depression are 11o and 8o respectively, how far apart are the merry-go-round and the Ferris wheel?

Trigonometry - J



7. Kirk visits Yellowstone Park and Old Faithful on a perfect day. His eyes are 6 feet from the ground and the geyser can reach heights ranging from 90 feet to 184 feet. A) If Kirk stands 200 feet from the geyser and the eruption rises 175 feet in the air, what is the angle of elevation to the top of the spray to the nearest tenth? B) In the afternoon, Kirk returns and observes the geyser’s spray reach a height of 123 feet when the angle of elevation is 37o. How far from the geyser is Kirk? 8. On July 20, 1969, Neil Armstrong became the first human to walk on the moon. During this mission, the lunar lander Eagle traveled aboard Apollo II. Before sending Eagle to the surface of the moon, Apollo 11 orbited the moon 3 miles above the surface. At one point, the onboard guidance system measured the angle of depression to the far and near edges of a large crater. The angle measured 16o and 29o, respectively. Find the distance across the crater.

Trigonometry - J



A Variety of Problems Answer Key

Draw a diagram and solve each of the following problems using trigonometric properties. 1. Joshua is flying a kite with 135 feet of string out. The angle of elevation of the kite varies from 45 degrees to 60 degrees during the flying time. What is the range of heights the kite will fly during this time? 95.46 feet to 116.91 feet 2. A jet plane is descending from 6000 feet above ground into DFW airport. The angle of depression between the pilot’s line of sight and the control tower is 5o. How many miles is the plane from the control tower? 68580.31 feet = 12.99 miles 3. One of the tallest mountains in Alaska is 15,308 feet above sea level, just 14.25 miles from the Pacific coast. What is the angle of elevation from the summit of the mountain to the shoreline of the Pacific Coast? 11.5o

Trigonometry - J



4. What is the measure of the angle of depression from a tower that is 120 feet tall to a point that is 525 feet away from the tower on level ground? 12.9o 5. A golfer is standing at the tee, looking up to the green on a hill. If the tee is 36 yards lower than the green and the angle of elevation from the tee to the hole is 12o. Find the distance from the tee to the hole. 173.2 yards 6. From the top of a roller coaster, 60 yards above the ground, a rider looks down and sees the merry-go-round and Ferris wheel. If the angles of depression are 11o and 8o respectively, how far apart are the merry-go-round and the Ferris wheel? 118.2 yards

Trigonometry - J



7. Kirk visits Yellowstone Park and Old Faithful on a perfect day. His eyes are 6 feet from the ground and the geyser can reach heights ranging from 90 feet to 184 feet. A) If Kirk stands 200 feet from the geyser and the eruption rises 175 feet in the air, what is the angle of elevation to the top of the spray to the nearest tenth? 40.2o B) In the afternoon, Kirk returns and observes the geyser’s spray reach a height of 123 feet when the angle of elevation is 37o. How far from the geyser is Kirk? 155.3 feet 8. On July 20, 1969, Neil Armstrong became the first human to walk on the moon. During this mission, the lunar lander Eagle traveled aboard Apollo II. Before sending Eagle to the surface of the moon, Apollo 11 orbited the moon 3 miles above the surface. At one point, the onboard guidance system measured the angle of depression to the far and near edges of a large crater. The angle measured 16o and 29o, respectively. Find the distance across the crater. 5.1 miles

Trigonometry - J



The Big Wheel

Name Megan's little brother Andrew has a bicycle with training wheels. The bicycle wheels have a 24 inch diameter and the training wheels have a 6 inch diameter. Andrew is riding at a steady pace and the big wheels rotate once every 4 seconds. The training wheels have a 6 inch diameter. The horizontal distance between the place where the front wheel touches the ground and where the training wheel touches the ground is 12π inches. As Andrew is riding down the street, he crosses a freshly painted stripe on the road. Answer the following questions to find out when the paint on the front tire will be the same height as the paint on the training wheel and what that height will be?

1. How much time elapses between when the front wheel crosses the stripe and the training wheel crosses the stripe?

2. Complete the chart. Do not fill in the gray spaces.

Time elapsed (in sec)

Height (in inches)of paint on front wheel

Height (in inches) of paint on training wheel




0 2.25 .25 2.5 .5 2.75 .75 3 1 3.25 1.25 3.5 1.5 3.75 1.75 4 2 4.25

3. What is the period of rotation of the front wheel? Use t=0 as the time that the front wheel crosses the stripe. When will the height of the paint first be one-half of its maximum displacement?

4. What is the period of rotation of the training wheel? Use t=0 as the time that the front wheel crosses the stripe. When is the first time that the height of the paint will be one-half of its maximum displacement?

Trigonometry - J



5. Use t=0 as the time that the front wheel crosses the stripe. Use the sine function to write

the height of the paint above the stripe as a function of time.

6. Remember that the training wheel reaches the stripe later than the front wheel. Write the height of the paint on the training wheel as a function of time, still using t=0 as the time that the front wheel crosses the stripe.

7. Graph the functions on your graphing calculator and find the first time that the paint is at the same height on both wheels and what that height is.

Trigonometry - J



The Big Wheel Answer Key

Megan's little brother Andrew has a bicycle with training wheels. The bicycle wheels have a 24 inch diameter and the training wheels have a 6 inch diameter. Andrew is riding at a steady pace and the big wheels rotate once every 4 seconds. The training wheels have a 6 inch diameter. The horizontal distance between the place where the front wheel touches the ground and where the training wheel touches the ground is 12π inches. As Andrew is riding down the street, he crosses a freshly painted stripe on the road. Answer the following questions to find out when the paint on the front tire will be the same height as the paint on the training wheel and what that height will be?

1. How much time elapses between when the front wheel crosses the stripe and the training wheel crosses the stripe?

2. Complete the chart. Do not fill in the gray spaces.







0 0 2.25 3 .25 2.5 0 .5 0 2.75 3 .75 3 3 12 6 1 12 6 3.25 3 1.25 3 3.5 0 1.5 0 3.75 3 1.75 3 4 0 6 2 24 6 4.25 3

3. What is the period of rotation of the front wheel? Use t=0 as the time that the front wheel crosses the stripe. When will the height of the paint first be one-half of its maximum displacement?

Period is 4. At 1 second

4. What is the period of rotation of the training wheel? Use t=0 as the time that the front wheel crosses the stripe. When is the first time that the height of the paint will be one-half of its maximum displacement?

Period is 1. at .75 seconds

Trigonometry - J



5. Use t=0 as the time that the front wheel crosses the stripe. Use the sine function to write

the height of the paint above the stripe as a function of time.

212sin( ( 1)) 12πy x= − +

6. Remember that the training wheel reaches the stripe later than the front wheel. Write the height of the paint on the training wheel as a function of time, still using t=0 as the time that the front wheel crosses the stripe.

3sin(2 ( 1.75)) 3y π x= − +

7. Graph the functions on your graphing calculator and find the first time that the paint is at the same height on both wheels and the height at that time.

After 3.67 seconds, the height on each will be 1.57 inches.

Trigonometry - J



Find the Missing Side or Angle Name Instructions: Find the missing side or angle as indicated in each of the right triangles below. 1. 2. x = ___________ θ = ___________ 3. 4. x = ___________ c = ___________ 5. 6. θ = ___________ x = __________ 7. 8. a = ___________ b = __________ 9. Describe a situation when you would use sine. Use illustrations to support your answer. 10. Describe a situation when you would use cos-1. Use illustrations to support your answer.

Trigonometry - Reteach

10 28°

x 23 18

θ

9

70°

x 30 45°

c

11

θ

2

25

55°x

17

65° a 8

38°

b



Find the Missing Side or Angle Answer Key

Instructions: Find the missing side or angle as indicated in each of the right triangles below. 1. 2. x = 18.81 θ = 51.5o 3. 4. x = 8.46 c = 21.21 6. 6. θ = 10.30 x = 14.34 7. 8. a = 7.93 b = 6.25 9. Describe a situation when you would use sine. Use illustrations to support your answer. When you know the measure of an angle and the measure of either the opposite side or the

hypotenuse.

10. Describe a situation when you would use cos-1. Use illustrations to support your answer. When you know the measure of the adjacent side and the hypotenuse and want to find the

measure of the angle.


10 28°

x 23 18

θ

9

70°

x 30 45°

c

11

θ

2

25

55°x

17

65° a 8

38°

b

15 25°

x

10

5 x°



Paper Plate

Name The unit circle provides a simple way to remember and calculate the trigonometric ratios of the special right triangles. In fact, you only need to remember the following: • the hypotenuse is always 1,

• the legs of the 45°, 45°, 90° have a measure of 22 ,

• the legs of the 30°, 60°, 90° have measures of 12 and 3

2 ,

• the side opposite the 30° angle = 12 ,

• and the side opposite the 60° angle = 32 .

Instructions: 1. Fold the paper plate into fourths and then fold in half. Open the paper plate and mark the 45°

angles. Label each leg 22 . Whether the measure is positive or negative is determined by the

quadrant in which it lies. 2. Fold the paper plate into fourths again and then into thirds. Open the paper plate and mark the 30°, 60°, 90° angles and corresponding sides, remembering that the side across from the 30°

angle measures 12 and the side across from the 60° angle measures 3

2 . The quadrant

determines whether the measure is positive or negative. When complete your plate will look like this:




Crop Circles

Name Marti the Martian needs your help! He has a deadline to meet on his crop circle project and doesn’t have time to complete the drawn plans to submit to his boss. Instructions: • Help Marti by drawing the circles or partial circles in the boxes provided. Knowing that you

may not speak Martian, Marti put the specifications in terms of the unit circle. • Start at 0° and move in the direction indicated until you reach the place on the circle equal to

the given cosine and sine values. Shade the portion of the circle bounded by the arc you just drew.

Example: Clockwise, cos θ = 1- 2 , sin θ = 3- 2

Step 1 Step 2

1. Counter-clockwise, cos θ = 2- 2 , sin θ = 2- 2 2. Clockwise, cos θ = 22 , sin θ = 2

2

3. Counter-clockwise, cos θ = 32 , sin θ = 1- 2 4. Clockwise, cos θ = 1

2 , sin θ = 32

Start here

0°

Move clockwise

Find the place on the circle for the given cos and sin and shade back to 0°.


Move clockwise

0°

1 3- , -2 2⎛ ⎞⎜ ⎟⎝ ⎠



5. Clockwise, cos θ = 3- 2 , sin θ = 12 6. Clockwise, cos θ = 0 , sin θ = 1

7. Counter-clockwise, cos θ = -1, sin θ = 0 8. Clockwise, cos θ = 1, sin θ = 0

9. Counter-clockwise, cos θ = 0, sin θ = -1 10. Clockwise, cos θ = 32 , sin θ = 1

2

11. Give the exact value of the following (exact values are expressed as integers or fractions and

may include radicals; if there is no value, write undefined):

cos 30° = sin 30° = tan 30° = cos 60° = sin 60° = tan 60° = cos 45° = sin 45° = tan 45° = cos 90° = sin 90° = tan 90° = cos 0° = sin 0° = tan 0° =

Thanks for your help!




Crop Circles Answer Key

Marti the Martian needs your help! He has a deadline to meet on his crop circle project and doesn’t have time to complete the drawn plans to submit to his boss. Instructions: • Help Marti by drawing the circles or partial circles in the boxes provided. Knowing that you

may not speak Martian, Marti put the specifications in terms of the unit circle. • Start at 0° and move in the direction indicated until you reach the place on the circle equal to

the given cosine and sine values. Shade the portion of the circle bounded by the arc you just drew.

Example: Clockwise, cos θ = 1- 2 , sin θ = 3- 2

Step 1 Step 2

1. Counter-clockwise, cos θ = 2- 2 , sin θ = 2- 2 2. Clockwise, cos θ = 22 , sin θ = 2

2

3. Counter-clockwise, cos θ = 32 , sin θ = 1- 2 4. Clockwise, cos θ = 1

2 , sin θ = 32


Start here

0°

Move clockwise

Find the place on the circle for the given cos and sin and shade back to 0°.

Move clockwise

0°

1 3- , -2 2⎛ ⎞⎜ ⎟⎝ ⎠



5. Clockwise, cos θ = 3- 2 , sin θ = 12 6. Clockwise, cos θ = 0 , sin θ = 1

7. Counter-clockwise, cos θ = -1, sin θ = 0 8. Clockwise, cos θ = 1, sin θ = 0

9. Counter-clockwise, cos θ = 0, sin θ = -1 10. Clockwise, cos θ = 32 , sin θ = 1

2

12. Give the exact value of the following (exact values are expressed as integers or fractions and

may include radicals; if there is no value, write undefined):

cos 30° = 32 sin 30° = 1

2 tan 30° = 1 3= 33

cos 60° = 12 sin 60° = 3

2 tan 60° = 3

cos 45° = 22 sin 45° = 2

2 tan 45° = 1

cos 90° = 0 sin 90° = 1 tan 90° = undefined cos 0° = 1 sin 0° = 0 tan 0° = 0

Thanks for your help!




Memory Match – Up

Students can be put into groups of 3 – 4. First place all cards face down and have each student take turns drawing two cards. If the two cards drawn go together as a pair, then the student will keep it as a match. The student with the most matches wins. Note: There are 3 cards that say “1”. There are 2 cards that say “ 3 ”. There are 2 cards that

say “ 32 ”. There are 2 cards that say “ 1

2 ”. Make sure that students know that two cards with

the exact same expression on them are not considered a match. For example: A card with a “1” on it does not match a card with a “1” on it. A card with a “1” on it is a match with a card that has “tan 45º” on it.




Memory Match – Up Cards

Pythagorean Theorem

2

1

1

3

45o

45o

?

45o

45o

?

?

30o

60o

?




3

12

22

1

tan 45º

sin 45º

sin 30º

cos 30º




It can be used to solve for an acute angle in a right triangle.

12

32

32

sin -1 33

30o

60o

?

30o

60o

?




sin θ cos θ

tan θ

opposite hypotenuse

adjacent

hypotenuse

opposite adjacent

leg2 + leg2 = hypotenuse2

2




tan 30º

sin 60º

cos 60º tan 60º




Memory Match-Up Answer Key

Pythagorean Theorem leg2 + leg2 = hypotenuse2

sin θ opposite hypotenuse

cos θ adjacent

hypotenuse

tan θ opposite adjacent

sin 30º 12

cos 30º 32

tan 30º 33

sin 45º 22

tan 45º 1 sin 60º 3

2

cos 60º 12

tan 60º 3

sin -1 It can be used to solve for an acute angle in a right triangle.

1

2

2

1

3

45º

45º

?

?

45º

45º ?

?

60º

30º

60º

30º

?

? 60º

30º




When Good Trig Goes Bad--In Search of Actual Triangles

Name Existence: answers the question “Can I find some object that has these properties?” Uniqueness: answers the question “Is there only one object with these properties?” 1. SSS (Side-Side-Side): A. The question: If we know three sides of a triangle, how many different triangles does this describe? Consider our three demonstration examples: 5, 12, 13 4, 5, 11 4, 11, 15 B. In your own words, why do only some combinations of sides yield a triangle? C. State the Triangle Inequality: D. If the side-lengths check out by triangle-inequality, how many triangles can we get with those exact sides? Here’s a step in the right direction: if you know the three sides of a triangle, what else can we determine about the triangle (and how do we do so)? E. From this, we may conclude that given for three sides that obey triangle inequality, exactly one triangle can be formed (i.e. there exists one unique triangle). 2. AAA (Angle-Angle-Angle) A. If we know just the three angle-measures, how many triangles are possible? (Assumption: the given angles add up to ______________ ). B. Let’s start with a familiar example--an equilateral triangle. What are the angle-measures of an equilateral triangle? C. How many triangles can we build with that exact combination of angle measures?

Trigonometry - Extension



D. For another example, let’s consider the Pythagorean triple 3-4-5 (the two legs and hypotenuse of a right triangle). What do you know about the triangle with sides 6-8-10? It is ______________________ to 3-4-5, meaning that: a. corresponding angles b. corresponding sides E. So, for a given angle-angle-angle combination, how many triangles are possible? 3. AAS (Angle-Angle-Side) or ASA (Angle-Side-Angle): A. To begin with, what can we find easily (without trig) in either case? B. From here, what law can we easily apply? C. Can you come up with any situation where we would not get a unique triangle determined? 4. SSA (Side-Side-Angle) A. Try to solve the following example: triangle ABC has a=5, b=8, and A=110º. Start by solving for the measure of B. What problems do we run into? B. Now consider this problem in more intuitive terms. Just like triangle-inequality, we have a well-formedness condition relating sides and angles of a triangle. Specifically: the largest angle must be opposite the largest side, and the smallest angle must be opposite the smallest side. How can we apply this to our last example? C. So, what conclusion can we reach here? Given 2 sides and a non-included angle, it is possible that they not form a triangle (i.e. some SSA cases give 0 triangles possible). Let’s examine the SSA cases more methodically: 1) A is obtuse: we can use this last observation: Can either of the other two angles be bigger than an obtuse angle? Thus, the bigger/smaller (circle one) of the two smaller sides must be opposite the obtuse angle for us to have a triangle. Otherwise, we get zero possible triangles. 2) A is right. Again, can either of the remaining two angles be bigger than 90º? So this case behaves just like our obtuse example. If the largest side is opposite the right-angle, we get one triangle. Otherwise,we get 0 triangles possible.




3) A is acute. This is the most complicated scenario, so let’s pick it apart. To diagram this, always put the acute angle in the lower-left corner, and the 2 sides proceed clockwise from here (ASS). Draw a baseline, the angle, and the first side. How could we now find the height of this triangle? Now consider the possible outcomes, depending on the length of the second side (ASS): a. If this side is smaller than the height, describe what result we get? An example of this type: C=35º, c=3, b=8. Work through this example; how many triangles are possible? b. If this second side is equal to the height, then what do we know about the exact location of that second side (where must it lie)? Do we get any triangles, and if so, how many? We can illustrate this with: B=42º, b=6.69, a=10. Work through this example to verify your claims. c. If this second side is greater than the height, but smaller than the first side, where might this side lie in relation to the height? Do we get any triangles, and if so, how many? Try this example, and see how many triangles you can get with these properties: C=35º, b=8, c=6. d. If this second side is greater than the height, and also greater than (or equal to) the first side, where must this side lie in relation to the height? Do we get any triangles, and if so, how many? Try this example, and see how many triangles you can get with these properties: C=35º, b=8, c=11.




When Good Trig Goes Bad--In Search of Actual Triangles

Answer Key Existence: answers the question “Can I find some object that has these properties?” Uniqueness: answers the question “Is there only one object with these properties?” 1. SSS (Side-Side-Side): A. The question: If we know three sides of a triangle, how many different triangles does this describe? Consider our three demonstration examples: 5, 12, 13 4, 5, 11 4, 11, 15 B. In your own words, why do only some combinations of sides yield a triangle? Student answers will vary. When the first 2 sides are connected, the third must be able to “make it back” to the first vertex. C. State the Triangle Inequality: The sum of any 2 side-lengths must be greater than the third. D. If the side-lengths check out by triangle-inequality, how many triangles can we get with those exact sides? Here’s a step in the right direction: if you know the three sides of a triangle, what else can we determine about the triangle (and how do we do so)? Use the Law of Cosines to get any 1 angle. Then use Law of Sines or Law of Cosines to get a second angle. E. From this, we may conclude that given for three sides that obey triangle inequality, exactly one triangle can be formed (i.e. there exists one unique triangle). 2. AAA (Angle-Angle-Angle) A. If we know just the three angle-measures, how many triangles are possible? Infinite number (Assumption: the given angles add up to 180º ). B. Let’s start with a familiar example--an equilateral triangle. What are the angle-measures of an equilateral triangle? 60º-60º-60º C. How many triangles can we build with that exact combination of angle measures? Infinite number




D. For another example, let’s consider the Pythagorean triple 3-4-5 (the two legs and hypotenuse of a right triangle). What do you know about the triangle with sides 6-8-10? It is similar to 3-4-5, meaning that: a. corresponding angles are congruent b. corresponding sides are in proportion E. So, for a given angle-angle-angle combination, how many triangles are possible? Infinite Number

3. AAS (Angle-Angle-Side) or ASA (Angle-Side-Angle): A. To begin with, what can we find easily (without trig) in either case? Third Angle B. From here, what law can we easily apply? Law of Cosines C. Can you come up with any situation where we would not get a unique triangle determined? No 4. SSA (Side-Side-Angle) A. Try to solve the following example: triangle ABC has a=5, b=8, and A=110º. Start by solving for the measure of B. What problems do we run into?

Using Law of Sines =sin110 sin5 8

B , so sinB = 1.50351---not possible

B. Now consider this problem in more intuitive terms. Just like triangle-inequality, we have a well-formedness condition relating sides and angles of a triangle. Specifically: the largest angle must be opposite the largest side, and the smallest angle must be opposite the smallest side. How can we apply this to our last example? B would have to be bigger than 110º, but that’s not possible since a triangle’s angles total 180º. C. So, what conclusion can we reach here? Given 2 sides and a non-included angle, it is possible that they not form a triangle (i.e. some SSA cases give 0 triangles possible). Let’s examine the SSA cases more methodically: 1) A is obtuse: we can use this last observation: Can either of the other two angles be bigger than an obtuse angle? No; Bigger Thus, the bigger/smaller (circle one) of the two smaller sides must be opposite the obtuse angle for us to have a triangle. Otherwise, we get zero possible triangles. 2) A is right. Again, can either of the remaining two angles be bigger than 90º? No So this case behaves just like our obtuse example. If the largest side is opposite the right-angle, we get one triangle. Otherwise,we get 0 triangles possible. 3) A is acute. This is the most complicated scenario, so let’s pick it apart. Height = first-side*sine(angle)




To diagram this, always put the acute angle in the lower-left corner, and the 2 sides proceed clockwise from here (ASS). Draw a baseline, the angle, and the first side. How could we now find the height of this triangle? Now consider the possible outcomes, depending on the length of the second side (ASS): a. If this side is smaller than the height, describe what result we get? No triangles because the “second side” can’t make it back down to the base to complete the triangle. An example of this type: C=35º, c=3, b=8. Work through this example; how many triangles are possible? b. If this second side is equal to the height, then what do we know about the exact location of that second side (where must it lie)? Do we get any triangles, and if so, how many? It must lie on the altitude (we have a right triangle). One triangle results. We can illustrate this with: B=42º, b=6.69, a=10. Work through this example to verify your claims. c. If this second side is greater than the height, but smaller than the first side, where might this side lie in relation to the height? Do we get any triangles, and if so, how many? The second side could lie on either side of the altitude. Two possible triangles. Try this example, and see how many triangles you can get with these properties: C=35º, b=8, c=6. d. If this second side is greater than the height, and also greater than (or equal to) the first side, where must this side lie in relation to the height? Do we get any triangles, and if so, how many? To the right side of the altitude (or, outside the altitude). One triangle results. Try this example, and see how many triangles you can get with these properties: C=35º, b=8, c=11.

Trigonometry - I

PreCalculus Standards 3, 4 and 5 Noncartesian Representations


COLUMBUS PUBLIC SCHOOLS

MATHEMATICS CURRICULUM GUIDE

GRADE LEVEL STATE STANDARD 3, 4, and 5 TIME RANGE GRADING Pre-Calculus Geometry and Spatial Sense; Patterns, 35 - 40 days PERIOD Functions and Algebra; Data Analysis 4 and Probability

MATHEMATICAL TOPIC 5

Noncartesian Representations

CPS LEARNING GOALS A) Uses vectors to model and solve application problems. B) Uses parametric equations to model and solve application problems. C) Uses polar coordinates. D) Expresses complex numbers in trigonometric form and computes sums, differences, products, quotients, powers, and roots of complex numbers in trigonometric form.

COURSE LEVEL INDICATORS Course Level (i.e., How does a student demonstrate mastery?):

Performs vector addition and vector multiplication by a scalar. Determines if two or more vectors are parallel or perpendicular. Given a vector, creates vectors which are parallel or perpendicular to it. Uses trigonometric ratios to perform decomposition of vectors into its component horizontal

and vertical vectors. Converts between Cartesian and parametric representations. Uses a graphing calculator to model problems given in parametric format. Converts back and forth between Cartesian and polar coordinates. Graphs polar equations by hand and with a graphing calculator. Identifies the form and properties of the graph of a polar equation. Converts coordinates back and forth between trigonometric and Cartesian form. Uses a calculator to verify the nth roots of a given complex number.

Previous Level:

Uses basic trigonometric identities (sine, cosine, and tangent).



The description from the state, for the Number, Number Sense, and Operations says: Students demonstrate number sense, including an understanding of number systems and operations and how they relate to one another. Students compute fluently and make reasonable estimates using paper and pencil, technology-supported and mental methods. The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is: A. Demonstrate that vectors and matrices are systems having some of the properties as the real

number system. B. Develop an understanding of properties o and representations for addition and multiplication

of vectors and matrices. D. Demonstrate fluency in operations with real numbers, vectors, and matrices, using mental

computation or paper and pencil calculations for simple cases, and technology for more complicated cases.

E. Represent and compute with complex numbers.

The description from the state, for the Geometry and Spatial Sense Standard says: Students identify, classify, compare, and analyze characteristics, properties, and relationships of one-, two- and three-dimensional geometric figures, and objects. Students use spatial reasoning, properties of geometric objects, and transformations to analyze mathematical situations and solve problems.

The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is: B. Represent transformations within a coordinate system using vectors and matrices.

The description from the state, for the Patterns, Functions, and Algebra Standard says: Students use patterns, relations, and functions to model, represent, and analyze problem situations that involve variable quantities. Students analyze, model and solve problems using various representations such as tables, graphs, and equations. The grade-band benchmarks from the state, for this topic in the grade band 11 – 12 are: A. Analyze functions by investigating rates of change, intercepts, zeros, asymptotes and local

and global behavior. Analyze functions by investigating rates of change, intercepts, zeros, asymptotes and local and global behavior.

D. Apply algebraic methods to represent and generalize problem situations involving vectors and matrices.

The description from the state, for the Data Analysis and Probability Standard says: Students pose questions and collect, organize represent, interpret, and analyze data to answer those questions. Students develop and evaluate inferences, predictions, and arguments that are abased on data. The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is: D. Create and analyze tabular and graphical displays of data using appropriate tools, including

spreadsheets and graphing calculators.

The description from the state, for the Mathematical Processes Standard says: Students use mathematical processes and knowledge to solve problems. Students apply problem-solving and decision-making techniques, and communicate mathematical ideas. The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is: A. Enter Text here




If the forces 1 2 33, 4 , 2, 6 , and 1, 5F F F= = − = − − act together on point P, find the additional force required to create equilibrium. A) 4,7− B) 7, 4− C) 4, 7− D) 7, 4−

Which of the following vectors describes an 8 lb force acting in the direction of 3, 5u = − ? A) 8 3, 5−

B) 8 3, 534

−

C) 5 3, 534

−

D) 5 5,334

−

Noncartesian Representations – A




Answers/Rubrics



If the forces 1 2 33, 4 , 2, 6 , and 1, 5F F F= = − = − − act together on point P, find the additional force required to create equilibrium. A) 4,7− B) 7, 4− C) 4, 7− D) 7, 4−

Answer: A

Which of the following vectors describes an 8 lb force acting in the direction of 3, 5u = − ? A) 8 3, 5−

B) 8 3, 534

−

C) 5 3, 534

−

D) 5 5,334

−

Answer: B





Let u = <5, 8>. Find a vector v of form <x, y> such that u is perpendicular to v.

A 2200 pound car is parked on an inclined street, 14º from the horizontal. Find the magnitude of the force that must be applied to keep the car from rolling downhill. Find the force perpendicular to the street. Create a mathematical model (diagram) representing all information and show your work.




Answers/Rubrics



Let u = <5, 8>. Find a vector v of form <x, y> such that u is perpendicular to v.

A 2200 pound car is parked on an inclined street, 14º from the horizontal. Find the magnitude of the force that must be applied to keep the car from rolling downhill. Find the force perpendicular to the street. Create a mathematical model (diagram) representing all information and show your work. Solution: To keep the car from rolling downhill, we need to find v. Using

trigonometry, sin 14o = 2200

v .

v = 2200sin14o = 532.23 lb

The force perpendicular to the street is: cos14o = 2200

v .

v = 2200cos14o = 2134.65 lb A 4-point response gives a proper diagram and uses appropriate ratios to get correct answers. A 3-point response gives an accurate diagram, but makes one mistake in setting-up or calculating answers. A 2-point response includes a diagram, but has two errors in labeling or calculations. A 1-point response gives a properly labeled diagram but no set-up or calculations. A 0-point response shows no mathematical understanding of the task.

14º

2200 lb v

h




Which of the following describes the graph of the parametric equations 3 , 2 4 , 0x t y t t= + = − ≥ ?

A. a line B. a line segment C. a ray D. a parabola E. a circle

Which of the following points corresponds to t = –2 in the parameterization 2 63 , 8x t yt

= − = + ?

A. (–3, 8) B. (–12, 11) C. (12, 5) D. (–12, 5)

Noncartesian Representations – B




Answers/Rubrics Low Complexity Moderate Complexity


Which of the following describes the graph of the parametric equations 3 , 2 4 , 0x t y t t= + = − ≥ ?A. a line B. a line segment C. a ray D. a parabola E. a circle

Answer: C

Which of the following points corresponds to t = –2 in the parameterization 2 63 , 8x t yt

= − = + ?

A. (–3, 8) B. (–12, 11) C. (12, 5) D. (–12, 5)

Answer: D





A ball is thrown straight up from level ground with its position above ground at any time 0t ≥ given by x = 4, y = -16t2+80t+7. At what time will the rock be 88 ft. above the ground? A. 3.590 sec and 1.410 sec B. 3.50 sec C. 1.410 sec D. 3.50 and 7.1 sec

The parametric equations of flight on the moon are x = (v cos ϑ ) t and y = (v sin ϑ )t − 2.66t 2 . Use your graphing calculator to determine the approximate horizontal distance that a baseball tossed upward from the surface travels if it is thrown with an initial velocity of 68 feet per second at an angle of 14 degrees relative to the surface and the amount of time it remains above the surface. Show the equations entered, a sketch of the graph, and the window used. Explain how you used the graph to find the answer.




Answers/Rubrics

High Complexity

ANSWER: A Short Answer/Extended Response


A ball is thrown straight up from level ground with its position above ground at any time 0t ≥ given by x = 4, y = -16t2+80t+7. At what time will the rock be 88 ft. above the ground? A. 3.590 sec and 1.410 sec B. 3.50 sec C. 1.410 sec D. 3.50 and 7.1 sec

The parametric equations of flight on the moon are x = (v cos ϑ ) t and y = (v sin ϑ )t − 2.66t 2 . Use your graphing calculator to determine the approximate horizontal distance that a baseball tossed upward from the surface travels and the amount of time it remains above the surface if it is thrown with an initial velocity of 68 feet per second at an angle of 14 degrees relative to the surface. Show the equations entered, a sketch of the graph, the window used. Explain how you used the calculator to find the answer.

Solution: The object remains in the air for approximately 6.19 seconds and travels approximately 408.4 ft. The student may trace or use the table to find the answer. A 2-point response demonstrates appropriate use of the calculator and describes how the proper x-coordinate is found. A 1-point response describes the appropriate use of the calculator, but does not find the correct answer. A 0-point response shows no mathematical understanding of the task.

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Given a point in polar form 58,

6π⎛ ⎞

⎜ ⎟⎝ ⎠

, which of the following expressions can be used to convert

to Cartesian coordinates?

A) 5 58sin , 8cos6 6

x yπ π⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

B) 2

2 55 , 56

x yπ⎛ ⎞= + =⎜ ⎟⎝ ⎠

C) 2

2 58 ,6

x y π⎛ ⎞= = ⎜ ⎟⎝ ⎠

D) 5 58cos , 8sin6 6

x yπ π⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Which one of the following polar coordinate pairs represents the same point as the point with polar coordinates ( , )r θ− (assume that r does not equal 0)? A) ( , )r θ B) ( , )r θ π+ C) ( , )r π θ− D) ( , )r θ−

Noncartesian Representations – C






Given a point in polar form 58,6π⎛ ⎞

⎜ ⎟⎝ ⎠

, which of the following expressions can be used to convert

to Cartesian coordinates?

A) 5 58sin , 8cos6 6

x yπ π⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

B) 2

2 55 , 56

x yπ⎛ ⎞= + =⎜ ⎟⎝ ⎠

C) 2

2 58 ,6

x y π⎛ ⎞= = ⎜ ⎟⎝ ⎠

D) 5 58cos , 8sin6 6

x yπ π⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Answer: D

Which one of the following polar coordinate pairs represents the same point as the point with polar coordinates ( , )r θ− (assume that r does not equal 0)? A) ( , )r θ B) ( , )r θ π+ C) ( , )r π θ− D) ( , )r θ−

Answer: B




Noncartesian Representations– C

Which of the following gives a maximum r-value for the polar graph of 3 4sinr θ= − ? A) 9 B) 8 C) 7 D) 6

Convert the point (6, –3) from Cartesian to polar coordinates. Give four forms for your answer: positive r with positive θ , positive r with negative θ , negative r with positive θ , and negative r with negative θ .




Answers/Rubrics



Which of the following gives a maximum r-value for the polar graph of 3 4sinr θ= − ? A) 9 B) 8 C) 7 D) 6

Answer: C

Convert the point (6, –3) from Cartesian to polar coordinates. Give four forms for your answer: positive r with positive θ , positive r with negative θ , negative r with positive θ , and negative r with negative θ . Solution:

( )

( )

2 2

1 36

6 ( 3) 45 3 5

tan 26.6

r

θ − −

= + − = =

= = − °

four points: (3 5, 26.6 ),(3 5,333.4 ),( 3 5,153.4 ),( 3 5, 206.6 )− ° ° − ° − − ° A 2-point response correctly determines the radius, uses the tangent to determine the angle, and properly determines the other three forms. A 1-point response finds the correct radius and angle but makes at least one error in the other forms. A 0-point response shows no mathematical understanding of the task.




Which of the following gives the number of distinct solutions of 4 3 5z i= + ?

A. 0 B. 1 C. 2 D. 3 E. 4 F. 5

Which Cartesian coordinate has a trigonometric form of 3 35 cos sin2 2

iπ π⎛ ⎞+⎜ ⎟⎝ ⎠

?

A. (5, 0) B. (0, 5) C. (–5, 0) D. (0, –5) E. (0, –1)

Noncartesian Representations – D






Which of the following gives the number of distinct solutions of 4 3 5z i= + ? A. 0 B. 1 C. 2 D. 3 E. 4 F. 5

Answer: E

Which Cartesian coordinate has a trigonometric form of 3 35 cos sin2 2

iπ π⎛ ⎞+⎜ ⎟⎝ ⎠

?

A. (5, 0) B. (0, 5) C. (–5, 0) D. (0, –5) E. (0, –1)

Answer: D





What are the three cube roots of -1?

A. 1 3 1 3, 1, and -2 2 2 2

i i− − − +

B. 1 3 1 3, 1, and 2 2 2 2

i i− +

C. 1 3 1 3, 1, and -2 2 2 2

i i− − +

D. 1 3 1 3, 1, and 2 2 2 2

i i− − +

Using DeMoivre’s Theorem, determine the 3 cube-roots of 2 + 2i in standard form (a + bi).




Answers/Rubrics

High Complexity What are the three cube roots of -1?

A. 1 3 1 3, 1, and -2 2 2 2

i i− − − +

B. 1 3 1 3, 1, and 2 2 2 2

i i− +

C. 1 3 1 3, 1, and -2 2 2 2

i i− − +

D. 1 3 1 3, 1, and 2 2 2 2

i i− − +

ANSWER: D Short Answer/Extended Response

Noncartesian Representations– D

Using DeMoivre’s Theorem, determine the 3 cube-roots of 2 + 2i in standard form (a + bi). Solution: for 2 + 2i, 2 22 2 2 2r = + = ( )1 2

2tan 45θ −= = °

( ) ( )11361 8,45 8 ,15 1.366 .366r i= ° = = +

( ) ( )11362 8,45 360 8 ,135 1r i= + = = − +

( ) ( )11363 8,45 720 8 , 255 .366 1.366r i= + = = − −

A 4-point response correctly applies DeMoivre’s Theorem to find the roots and expresses them in standard form. A 3-point response properly applies DeMoivre’s Theorem to find the roots and expresses them in standard form with only one slight error A 2-point response carries out the conversion into polar form properly and sets up the DeMoivre’s Theorem appropriately but fails to put answers in standard form. A 1-point response carries out the conversion into polar form properly A 0-point response shows no mathematical understanding of the problem.



Teacher Introduction NonCartesian Representations

Although this topic includes many avenues of study, this course offers only an introduction into vectors, parametrics, polar equations, and the trigonometric form of complex numbers. If time allows, the text offers much greater depth into these areas of study. Vectors are generally included in a high school physics course and would probably be studied in depth there. They are included here to provide the mathematical background necessary to understand them and to provide an introduction for students who do not study physics. Parametrics offer the ability to graph relations which are not functions and to look at graphs which are functions differently. The use of a parameter enables many real world applications where there is a controlling factor, usually time, which is the basis for the creation of the graph. Linear and quadratic parametric equations could easily be included in the earlier studies of lines and parabolas. The students should understand the basis of polar graphing, the pair consisting of a radius and an angles, and should be able to convert back and forth between polar and Cartesian coordinates. They should graph simple polar equations by hand and use a graphing calculator to graph complicated polar equations. Students should recognize rose curves, limaçon curves, cardioids, and lemniscate curves.



TEACHING STRATEGIES/ACTIVITIES

Core: Learning Goal A: Uses vectors to model and solve application problems. 1. Use the activity Current Swimming, included in this curriculum guide, to introduce

separating a vector into its components and adding vectors by adding components. Save this activity for use later in the topic.

Learning Goal B: Uses parametric equations to model and solve application problems. 1. Use the activity Ships, included in this curriculum guide, to introduce parametric graphing. 2. Use the activity Parametric Swimming to emphasize the relationship between the component

form of vectors and parametrics. Learning Goal C: Uses polar coordinates. 1. Have students do the activity Polar Flora, included in this curriculum guide to use the

graphing calculator to investigate polar graphing.. Learning Goal D: Expresses complex numbers in trigonometric form and computes sums, differences, products, quotients, powers, and roots of complex numbers in trigonometric form. 1. Complete the exercise DeMoivre vs. Factoring to provide the connection between the

trigonometric form of roots and the more familiar quadratic formula.

RESOURCES Learning Goal A: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 502-521 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp. 117-120 Learning Goal B: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 522-533 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp. 121-122 Learning Goal C: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 534-549 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp. 123-126

Vocabulary: vector, component, magnitude, scalar unit vector, dot product, projection, parameter, polar, trigonometric form, DeMoivre's Theorem



Learning Goal D: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 550-557 Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp. 127-128



Current Swimming

Name Mary, Tamika, and Gabriella are in a contest to see who can swim across a river the fastest. Each of them can swim at the rate of .2 m/sec in still water. The river is 50 meters wide and flows from north to south at .1 m/sec. The girls are going to start on the western bank of the river and swim to the eastern shore. Each girl decides upon a strategy to cross the river, bearing in mind that the river is going to affect the path. Mary decides to head straight across the river. Tamika decides to head 30º N of E. Gabriella decides to head 30º S of E. Who will win the race? Mary

1. If there were no current, Mary would head straight across the river. The current moves her moves her.1 m due south. Complete the chart for her position, using (0,0) for her initial position. Make a sketch illustrating her motion.

Time

(in seconds) x-position (in meters)

y-position (in meters)

without current

effect of current on y-position

y-position (in meters) with current

1 10 20 50 100

t

2. How many seconds does it take Tamika to cross the river?

3. Does she travel upstream, downstream or straight across the river?

4. How far upstream or downstream does she end up?

5. What is her displacement, the distance between her initial and final positions?

6. What are her speed and direction?

Vectors – A



Tamika

7. If Tamika heads 30º N of E, draw a sketch of her path and use trigonometry to describe her motion to the east and north.

8. Complete the chart for her position, using (0,0) for her initial position. Make a sketch illustrating her motion.

Time



without current

effect of current on y-

position


1 10 20 50 100

t

9. How many seconds does it take Tamika to cross the river?





Vectors – A



Gabriella

14. If Gabriella heads 30º S of E, draw a sketch of her path and use trigonometry to describe her motion to the east and north.


Time (in seconds) x-position (in meters) y-position (in meters)

without current y-position (in meters)

with current 1 10 20 50 100

t

16. How many seconds does it take Gabriella to cross the river?





21. All the girls would be swimming the same speed in still water. Who is actually moving the fastest?, the slowest? Who reaches the eastern bank first?

Vectors – A



Current Swimming Answer Key

Name Mary, Tamika, and Gabriella are in a contest to see who can swim across a river the fastest. Each of them can swim at the rate of .2 m/sec in still water. The river is 50 meters wide and flows from north to south at .1 m/sec. The girls are going to start on the western bank of the river and swim to the eastern shore. Each girl decides upon a strategy to cross the river, bearing in mind that the river is going to affect the path. Mary decides to head straight across the river. Tamika decides to head 30º N of E. Gabriella decides to head 30º S of E. Who will win the race? Mary

1. If there were no current, Mary would head straight across the river. The current moves her moves her.1 m due south. Complete the chart for her position, using (0,0) for her initial position. Make a sketch illustrating her motion.

Time

(in seconds) x-position

(in meters) y-position (in meters)

without current

effect of water on y-position


1 .2 0 -.1 -.1 10 2 0 -1 -1 20 4 0 -2 -2 50 10 0 -5 -5 100 20 0 -10 -10

t .2t 0 -.1t 0--.1t

1. How many seconds does it take Mary to cross the river? 250 sec

2. Does she travel upstream, downstream or straight across the river? Downstream

3. How far upstream or downstream does she end up? 25 m

4. What is her displacement, the distance between her initial and final positions? 55.902 m

5. What are her speed and direction? .223 m/s 26.565º Sof E

Vectors – A



Tamika

6. If Tamika heads 30º N of E, draw a sketch of her path and use trigonometry to describe her motion to the east and north.


Time



without current

effect of current on y-position


1 .2 cos 30º=.173 .1 -.1 0 10 2 cos

30º=1.732 1 -1 0

20 4 cos 30º=3.4641

2 -2 0

50 10 cos 30º=8.660

5 -5 0

100 20 cos 30º=17.321

10 -10 0

t .2t cos 30º .1t -.1t .1t+.1t=0

8. How many seconds does it take Tamika to cross the river? 288.675 sec

9. Does she travel upstream, downstream or straight across the river? straight across

10. How far upstream or downstream does she end up? 0 meters

11. What is her displacement, the distance between her initial and final positions? 50 meters

12. What are her speed and direction? .173 m/sec, 0º

.2 .2sin30 .2 cos 30

Vectors – A



Gabriella

13. If Gabriella heads 30º S of E, draw a sketch of her path and use trigonometry to describe her motion to the east and north.


Time



without current

Effect of current on y-

position


1 .2 cos 30º=.173 -.1 -.1 -.2 10 2 cos 30º=1.732 -1 -1 -2 20 4 cos 30º=3.4641 -2 -2 -4 50 10 cos 30º=8.660 -5 -5 -10 100 20 cos 30º=17.321 -10 -10 -20

t .2t cos 30º -.1t -.1t -.1t-.1t-.2t

15. How many seconds does it take Gabriella to cross the river? 288.675 sec

16. Does she travel upstream, downstream or straight across the river? Downstream

17. How far upstream or downstream does she end up? 57.735 m

18. What is her displacement, the distance between her initial and final positions? 76.376 m

19. What are her speed and direction? .265 m/s and 49.107º S of E

20. All the girls would be swimming the same speed in still water. Who is actually moving the fastest?, the slowest? Who reaches the eastern bank first?

Fastest Gabriella Slowest Tamika Mary reaches the eastern bank first.

.2cos 30º .2sin30 .2

Vectors – A



Ships

Name You are watching a radar screen that is a square with sides 1000 mm in length. Two ships, the Minnow and the Pinafore, appear on your radar screen. As they come onto your screen, the Minnow is at a point 900 mm from the bottom left corner of the screen along the lower edge. The Pinafore is located at a point 100 mm above the lower left corner along the left edge. One minute later the positions have changed. The Minnow has moved to a location on the screen that is 3mm W and 2 mm N of its previous location. The Pinafore has moved 4mm E and 1 mm N. They continue to move at constant speeds on their respective linear courses. WILL THE MINNOW AND THE PINAFORE COLLIDE? 1. Use the table to compute the position of each ship on the radar screen at the times indicated. In the final row, write an expression in terms of t that would provide a general description of the position at any time t.

Minnow Pinafore Time (t) x y. x y

0 1 2 3 10 25 t

2. Write an equation for the path of the Minnow. What are the slope, x- and y-intercepts? Write an equation for the path of the Pinafore. What are the slope, x- and y-intercepts? How do the slope, and the x- and y-intercepts relate to the motion? 3. Set the graphing mode of your calculator to Simultaneous. Graph the equations on your calculator using the window [0,1000] by [0,1000]. Where do the lines cross? Does this mean that the ships collide at that point? How can you be sure? 4. Use the expressions you wrote for the x and y coordinates of each ship to determine the time that each reaches the point of intersection of the paths.

Parametric – B



Minnow Pinafore Do they arrive at the point of intersection at the same time? 5. Set the mode of your calculator to Parametric. Look at the y= menu. Notice that there are now pairs of entries, one for x and one for y. Enter the expressions from the last row of the chart for the Minnow in X1T and Y1T and the Pinafore in X2T and Y2T. You can enter the T by using the key you use for x when in the function mode. When you select window, the x and y settings should still be the same as when you graphed the lines in #3. The t is called the parameter and is controlling x and y. Set TMIN at 0 and TMAX 10. Set TSTEP at 1. Hit graph. Change values for TMAX and TSTEP to see how they affect the graph. What do TMIN and TMAX do? What does TSTEP do? How does the parametric graph help you answer the question, "WILL THE SHIPS COLLIDE? 6. Two other ships, the Enterprise and Falcon, were spotted on a screen. The parametric

equations describing their paths were 1 2 51 3 10

X T tY T t

= −= +

for the Enterprise and 2 152 4 10

X T tY T t

= += −

for the

Falcon. Use parametric equations to simulate their motion, adjusting you window as needed. (They might not be visible on the radar screen at all times.) Do you think they collide? Show the math that proves whether or not they collide. What if these were space ships being sighted from the ground. Could you still be sure that they collide? Explain.

Parametric – B



7. What is the slope of the function that describes the path of the Enterprise? Give a point on the path of the Enterprise.. Write the function that describes the path of the Enterprise. What is the slope of the function that describes the path of the Falcon? Give a point on the path of the Falcon.. Write the function that describes the path of the Falcon.

8. The path of another ship is modeled by the equation 2 103

y x= − + . Write a pair of parametric

equations that describe this same line. 7. A rock is dropped from a tower that is 1000 ft. tall. At any time t measured in seconds, its distance from the ground in feet is 21000 16s t= − . You cannot graph the path of the rock using the function grapher on your calculator because it is a vertical line. Use the parametric mode to graph the path of the rock. Give your equations here.

Parametric – B



Ships Answer Key

You are watching a radar screen that is a square with sides 1000 mm in length. Two ships, the Minnow and the Pinafore, appear on your radar screen. As they come onto your screen, the Minnow is at a point 900 mm from the bottom left corner of the screen along the lower edge. The Pinafore is located at a point 100 mm above the lower left corner along the left edge. One minute later the positions have changed. The Minnow has moved to a location on the screen that is 3mm W and 2 mm N of its previous location. The Pinafore has moved 4mm E and 1 mm N. They continue to move at constant speeds on their respective linear courses. WILL THE MINNOW AND THE PINAFORE COLLIDE? 1. Use the table to compute the position of each ship on the radar screen at the times indicated. In the final row, write an expression in terms of t that would provide a general description of the position at any time t.

Minnow Pinafore Time (t) x y. x y

0 900 0 0 100 1 897 2 101 2 894 8 102 3 891 12 103 10 870 6 40 110 25 825 20 100 125 t

900-3t 50 4t 100+t

2. Write an equation for the path of the Minnow. What are the slope, x- and y-intercepts? y=-2/3x +600 Write an equation for the path of the Pinafore. What are the slope, x- and y-intercepts? y=-1/4x +100 How do the slope, and the x- and y-intercepts relate to the motion? The slope indicates the direction of the motion. The y-intercept does not relate to motion, it just indicates a particular position. 3. Set the graphing mode of your calculator to Simultaneous. Graph the equations on your calculator using the window [0,1000] by [0,1000]. Where do the lines cross? Does this mean that the ships collide at that point? How can you be sure? (545.455, 236.364) Answers will vary 4. Use the expressions you wrote for the x and y coordinates of each ship to determine the time that each reaches the point of intersection of the paths.

Parametric – B



Minnow Pinafore 118.182 sec 136.364 sec Do they arrive at the point of intersection at the same time? NO 5. Set the mode of your calculator to Parametric. Look at the y= menu. Notice that there are now pairs of entries, one for x and one for y. Enter the expressions from the last row of the chart for the Minnow in X1T and Y1T and the Pinafore in X2T and Y2T. You can enter the T by using the key you use for x when in the function mode. When you select window, the x and y settings should still be the same as when you graphed the lines in #3. The t is called the parameter and is controlling x and y. Set TMIN at 0 and TMAX 10. Set TSTEP at 1. Hit graph. Change values for TMAX and TSTEP to see how they affect the graph. What do TMIN and TMAX do? They give the starting and stopping time. What does TSTEP do? It gives how much the time increases from one point to the next. How does the parametric graph help you answer the question, "WILL THE SHIPS COLLIDE? You can see where they are at a given time. 6. Two other ships, the Enterprise and Falcon, were spotted on a screen. The parametric

equations describing their paths were 1 2 51 3 10

X T tY T t

= −= +

for the Enterprise and 2 152 4 10

X T tY T t

= += −

for the

Falcon. Use parametric equations to simulate their motion, adjusting you window as needed. (They might not be visible on the radar screen at all times.) Do you think they collide? Show the math that proves whether or not they collide. The window TMin 0, TMax 30, [0,100], [0,100] works well. The ships appear to collide. Setting X1T=X2T gives t=20. Substituting 20 into Y1T and Y2T gives 70 in both cases. This means that the ships were in the same place at this time and they do collide. What if these were space ships being sighted from the ground. Could you still be sure that they collide? Explain. No, they could be in different planes.

Parametric – B



7. What is the slope of the function that describes the path of the Enterprise? Give a point on the path of the Enterprise.. Write the function that describes the path of the Enterprise. What is the slope of the function that describes the path of the Falcon? Give a point on the path of the Falcon.. Write the function that describes the path of the Falcon. Enterprise slope is 3/2. y=3/2x + 17.5 Falcon slope is 4. y=4x –70

8. The path of another ship is modeled by the equation 2 103

y x= − + . Write a pair of parametric

equations that describe this same line. x=t, y=-2t + 10 7. A rock is dropped from a tower that is 1000 ft. tall. At any time t measured in seconds, its distance from the ground in feet is 21000 16s t= − . You cannot graph the path of the rock using the function grapher on your calculator because it is a vertical line. Use the parametric mode to graph the path of the rock. Give your equations here. x=5, y=1000 – 16t2. The number with x is arbitrary. It gives the equation of the vertical line the rocket moves along.

Parametric – B



Parametric Swimming

Name

Go back to the Activity Current Swimming. In this activity, you looked at the vectors for the paths of Mary, Tamika, and Gabriella as they swam across a river and created tables that described their motion and the effect of the river on their motion. To illustrate the motion on your calculator, enter this window:

In Format, turn the axes off. In Mode, choose Parametric and Simultaneous For each swimmer, enter the formula for her x-position in X1T. Enter the formula for her y-position with no current in Y1T. Enter 0 in X2T. Enter the formula for the current in Y2T In X3T, enter X1T+X2T. In Y3T, enter Y1T+Y2T

Sketch the diagram for each swimmer below. Identify the vector for the swimmer with no current, the current, and the swimmer in the current.

Mary Tamika Gabriella

Tmin= 0 Tmax= 250 Tstep= 1 Xmin= -10 Xmax= 50 Xscl= 5 Ymin= -30 Ymax= 30 Yscl= 5

Parametric – B



Parametric Swimming Answer Key

Go back to the Activity Current Swimming. In this activity, you looked at the vectors for the paths of Mary, Tamika, and Gabriella as they swam across a river and created tables that described their motion and the effect of the river on their motion. To illustrate the motion on your calculator, enter this window:

In Format, turn the axes off. In Mode, choose Parametric and Simultaneous For each swimmer, enter the formula for her x-position in X1T. Enter the formula for her y-position with no current in Y1T. Enter 0 in X2T. Enter the formula for the current in Y2T In X3T, enter X1T+X2T. In Y3T, enter Y1T+Y2T

Sketch the diagram for each swimmer below. Identify the vector for the swimmer with no current, the current, and the swimmer in the current. Mary Tamika Gabriella

Tmin= 0 Tmax= 290 Tstep= 1 Xmin= -10 Xmax= 50 Xscl= 5 Ymin= -30 Ymax= 30 Yscl= 5

Parametric – B



Polar Flora

Name

Use your calculator to graph each of the polar equations. Sketch each graph and indicate the window settings used to get a complete graph. Try to select the smallest interval possible between θmin and θmax. Give the length of each petal of the rose graph. For each equation, change sine to cosine (or cosine to sine) and see how the graph changes. Sketch How did the graph change? y=2 sin 3θ θ y=3 sin 2θ y=3 cos 2θ y=5 cos 4θ y=5 sin 4θ y=cos 5θ y=sin 5θ Window settings y=2 sin 3θ

y=2 cos 3 y=3 sin 2θ y=3 cos 2θ

y =5 cos 4θ y=5 sin 4θ

y=cos 5θ y=sin 5θ

θ min θ max θ step Xmin Xmax Ymin Ymax

Polar – C



1. What does the r-coordinate of each point represent?

2. What does the θ -coordinate of each point represent?

3. How do the choices of θ min, θ max, and θ step affect the graph?

4. How does the polar equation predict the length of each petal?

5. How does the polar equation predict the number of petals on the rose?

6. How did the graph change when you changed trig functions? Explain this in terms of the relation of the graphs of the sine and cosine function.

Polar – C



Use your calculator to graph each of the polar equations. Sketch each graph and indicate the window settings used to get a complete graph. Try to select the smallest interval possible between θmin and θmax. Give the length of each petal of the rose graph. For each equation, change sine to cosine (or cosine to sine) and see how the graph changes. Sketch How did the graph change? y=2 sin 3θ y=2cos 3θ The petals are in different positions.

y=3 sin 2θ y=3 cos 2θ

y=5 cos 4θ y=5 sin 4θ

y=cos 5θ y=sin 5θ

Polar Flora Answer Key

Window settingsAnswers are one possibility. Answers will vary. y=2 sin 3θ

y=2 cos 3 y=3 sin 2θ y=3 cos 2θ

y =5 cos 4θ y=5 sin 4θ

y=cos 5θ y=sin 5θ

θ min 0 0 0 0 θ max 2.10 3.1416 6.28 6.28 θ step .1308996 .1308996 .1308996 .1308996 Xmin -6.1522856 -6.1522856 -8 -2 Xmax 6.1522856 6.1522856 8 3 Ymin -4 -4 -6 -1.5 Ymax 4 4 6 1.5

Polar – C



1. What does the r-coordinate of each point represent? The distance from the pole.

2. What does the θ -coordinate of each point represent? The angle from the postive x-axis.

3. How do the choices of θ min, θ max, and θ step affect the graph? θ min and θ max determine whether the graph is completed or repeated.

4. How does the polar equation predict the length of each petal? The coefficient of the trig function determines the length of the petal.

5. How does the polar equation predict the number of petals on the rose? If the coefficient of θ, n, is even, then there are 2n petals. If odd, there are n petals

6. How did the graph change when you changed trig functions? Explain this in terms of the relation of the graphs of the sine and cosine function. The graphs are the same shape and size but in different places. This corresponds to the graphs of sine and cosine being out of phase.

Polar – C



DeMoivre vs. Factoring

Name

DeMoivre's Theorem is used to powers of complex numbers in trigonometric form and to find real and nonreal roots of complex numbers. If z=r(cos θ +i sin θ ), then DeMoivre's Theorem tells us that the n distinct complex numbers are

2 2(cos sin )n k kr in n

ϑ π ϑ π+ ++ , where k = 0, 1, 2, 3, 4, …, n – 1.

1. Write the trigonometric form of -1.

2. Use DeMoivre's Theorem to find the cube roots of -1. Express your answer in both

trigonometric and a+bi form.

3. The roots of -1 are the solutions to the equation x3+1=0. What one the obvious solution to the equation?

4. Use synthetic division to factor x3+1.

5. Use the quadratic formula to find the other two zeros.

6. Do the answers agree with the trigonometric formula?

7. Find the fourth root of a number using both of these methods.

Complex in Trig. Form – D



DeMoivre vs. Factoring Answer Key

DeMoivre's Theorem is used to powers of complex numbers in trigonometric form and to find real and nonreal roots of complex numbers. If z=r(cos θ +i sin θ ), then DeMoivre's Theorem tells us that the n distinct complex numbers are

2 2(cos sin )n k kr in n

ϑ π ϑ π+ ++ , where k = 0, 1, 2, 3, 4, …, n – 1.

1. Write the trigonometric form of -1. z = -1 + 0i

2. Use DeMoivre's Theorem to find the cube roots of -1. Express your answer in both

trigonometric and a+bi form.

1

2

3

1 3cos sin3 3 2 2

2 2cos sin 1 03 34 4 1 3cos sin

3 3 2 2

π π

π π π π

π π π π

= + = +

+ += + = − +

+ += + = −

z i i

z i i

z i i

3. The roots of -1 are the solutions to the equation x3+1=0. What one the obvious solution to the equation? -1

4. Use synthetic division to factor x3+1.

1 1 0 0 11 1 1

1 1 1 0

−− −

−

(x+1)(x2 +1)

5. Use the quadratic formula to find the other two zeros.

1

2

1 32 2

1 32 2

= +

== −

z i

z i

6. Do the answers agree with the trigonometric formula?

YES

7. Find the fourth root of a number using both of these methods. Answers will vary.

Complex in Trig. Form – D

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