marcolsen.weebly.com · Web viewWord Problems Practice 49 2.5 Further Applications of Right Triangles 51 7.1 Oblique Triangles and The Law of Sines 53 7.3 The Law of Cosines 59 7.2

Math 335

Math 335 Handouts

Section Topic Page NumberGraphing Review Functions and Circles 3-14

Appendices A, B and C Equations, Inequalities and Functions 151.1 Angles 171.2 Angle Relationships and Similar Triangles 21

3.1 (part I) Radian Measures 23Worksheet Warm-up 25

1.3 Trig Functions 271.4 Using Definitions of Trig Functions 312.1 Trig Functions of Acute Angles 352.2 Trig Functions of Non-acute Angles 39

3.1 (Part II) Radian Measures 393.2 Applications of Radian Measures 413.3 The Unit Circle and Circular Functions 432.4 Solving Right Triangles 47

Worksheet Word Problems Practice 492.5 Further Applications of Right Triangles 517.1 Oblique Triangles and The Law of Sines 537.3 The Law of Cosines 597.2 The Ambiguous Case of the Law of Sines 63

Summary Solving a Triangle 684.1 Sine and Cosine graphs 694.2 Translations of Sine and Cosine Graphs 754.3 Tangent and Cotangent Graphs 794.4 Secant and Cosecant Graphs 834.5 Harmonic Motion 85

worksheet Trig Graphing Review 875.1 Fundamental Identities 935.2 Verifying Trig Identities 955.3 Sum and Difference Identities for Cosine 995.4 Sum and Difference Identities for Sine and Tangent 1035.5 Double Angle Identities 107

Worksheet Identities Review 1115.6 Half Angle Identities 1156.1 Inverse Circular Functions 1196.2 Trig Equations I 127

Worksheet Factoring Review 1316.3 Trig Equations II 1336.4 Equations Involving Inverse Trig Functions 135

worksheet Solving Trig Equations Practice 137worksheet Exponential and logarithmic functions 139

7.4 Vectors, Operations and Dot Products 1417.5 Applications of Vectors (lecture examples) 1477.5 Applications of Vectors (exercises) 1498.1 Complex Numbers 1518.2 Polar Form of Complex Numbers 1558.3 The Product and Quotient Theorems 1598.4 The DeMoivre’s Theorem; Powers and Roots and Complex Numbers 1618.5 Polar Equations and Graphs 165

worksheet The Cycle of a Rose Curve 171worksheet Polar Equations and Graphs 173Homework Polar Equations and Graphs 177Templates Homework Forms Back of packet

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HW Form A (x4) - 16 copies HW Form B (x6) - 13 copies

HW Form C (graph) – 9 copies HW Form D (graph) – 5 copies

HW Form E (x3) - 26 copies HW Form F (x2) – 6 copies

Below is the list of all sections involving required problems and the type of HW form to be used with each.

Start each new section on a new form. Work in boxes LEFT TO RIGHT!

Section HW Form Section HW Form

App A A 5.6 E (x2)

App B D 6.1 A (x2)

App C A 6.2 E (x2)

1.1 B 6.3 E (x2)

1.2 B 6.4 A

3.1 (Part I) B exponential/logs worksheets

1.3 D & A 8.1 E

1.4 B 7.4 B (x2)

2.1 B 7.5 E

2.2 B 8.2 A

3.1 (part II) A 8.3 E

3.2 B 8.4 E

3.3 B 8.5 E (x2)

3.4 A

2.4 A

2.5 E

7.1 E (x2)

7.3 E (x2)

7.2 E (x2)

4.1 C (x2)

4.2 A & C

4.3 C (x2)

4.4 C

4.5 F

5.1 A

5.2 F (x2)

5.3 A

5.4 E (x2)

5.5 E (x2)

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Key Functions: Absolute Value

*Label three ordered pairs on the function’s graph on both sides of the vertex. (integer coordinates only)

*Use interval notation to list the following:

Domain:

Range:

I. Sketch each function on the given grid. List the coordinates for the vertex.

a)

b)

c)

d)

II. Evaluate and simplify each expression.

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a)

b)

c) Determine if the following statement is true or false. Show your work.

= ?

III. Solve each equation for all values of

‘x’ given that .

a)

b)

c)

IV. Write the function for each graph.

a)

b)

c)

Key Functions: Quadratic

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*Label seven ordered pairs on the function’s graph. (integer coordinates only)


Domain:

Range:

I. Sketch each parabola on the provide grids. List the coordinates of the vertex for each.

a)

b)

c)

d)

II. Evaluate and simplify each

expression given

a)

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b)

c)

d)


‘x’ given that . a)

b)


a)

b)

c)

Key Functions: Cubic

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*Label five ordered pairs on the function’s graph. (integer coordinates only)

*Use interval notation to list the function’s:

Domain:

Range:

I. Sketch each function on the given grid. List the coordinates of the critical point.

a)

b)

c)

d)


expression given that .

a)

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b)

c)

d)



a)

b)


a)

b)

c)

Key Functions: Rational

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*Label six ordered pairs on the function’s graph. (3 in each quadrant.)

Label the Asymptotes on the function’s graph.


Domain:

Range:

I. Sketch the graph on the given grid. Sketch the asymptotes with dotted lines and label them.

a)

b)

c)

d)



a)

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b)

c)



a)

b)

IV. Write the function for each graph.Draw and label the asymptotes.

a)

b)

c)

Key Functions: Square Root

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*Label four ordered pairs on the function’s graph. (integer coordinates only)


Domain:

Range:

I. Sketch each function on the given grid.

a)

b)

c)

d)



a)Page 11 of 186

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b)

c)



a)

b)

c)

IV. Write the function for each graph.a)

b)

c)

Circles _______________________

Circle: the set of points in a plane that are equidistant from a given point.

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List the center and radius of each circle.

1. center _________ Radius: r = _______

2. center _________ Radius: r = _______

3. center _________ Radius: r = _______

4. center _________ Radius: r = _______

5. center _________ Radius: r = _______

6. center _________ Radius: r = _______

7. center _________ Radius: r = _______

8. center _________ Radius: r = _______Graph each circle.

9. 10.

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11. 12.

Appendices A, B and C Class practice Name :_________________________Date :___________________

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1. Equations and Inequalities

Solve. Write your final answer using set notation or interval notation.

(a) (b)

(c)

2. Functions

(a) A relation is a set of ordered pairs.(b) A FUNCTION is a relation where ____________________________________________________.(c) Domain of a function is _____________________. (d) Range of a function is_________________.(e) 8 key functions studied in Intermediate Algebra:

Function Name General Equation Sketch of graph Domain and RangeLinear

Quadratic

Absolute Value

Cubic

Radical

Rational

Exponential

Logarithm

3. List the Domain of each function. (Use interval notation.)

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(a) (b) (c)

4. Circle all functions with a Range of all Real Numbers.

5. List the Domain and Range for each graph. (Use interval notation.)

D: _________________ R: ____________ D: ______________ R:______________

Problems 6 - 7: Use the list of functions to respond to each question.

6. Evaluate each of the following. Simplify if possible.

(a) (b)

(c) (d)

7. Solve for all values of x.

Chapter 1: Trigonometric FunctionsTrigonometry: The study of triangles.

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Sec 1.1: AnglesDefinitionsA line is an infinite set of points where between any two points, there is another point on the line that lies between them. Line AB, AB

A line segment is part of a line that consists of two distinct points on the line and all the points between them. Line segment AB or segment AB, AB

A ray AB, denoted AB. The endpoint of the ray AB is A. The endpoint of the ray BA is B.

An angle is a plane figure formed by two rays that share a common point (vertex).

Initial and terminal sides

vertex

Degree is a unit for measuring angles and arcs that corresponds to 1/360 of a complete revolution.

Counterclockwise rotation results in (+) measure

Clockwise rotation results (-) measure

Types of Angles MeasurementAcute angleRight angle

Obtuse angleStraight angle

Note: Angles are often denoted by the greek letter (theta). Other greek letters to represent angles are etc…..

DefnsComplementary angles are angles whose sum measures that add up to 90 ° (they’re complements of each other)Supplementary angles are angles whose sum measures that add up to 180 ° (they’re supplements of each other)

Ex 1Find the measure of the marked angles.

a = (4x + 12)°, b = (2x + 66)°Ex 2 Find the measure of the smaller angle formed by the hands of a clock at the given time.a) 4 :00

b) 1 :45

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initial side

terminal side

Math 335

What do we do when we have an angle that is less than 1 degree? Use FRACTIONS of DEGREES with minutes and seconds.

Note: Fractions of degrees are used in distance between two cities measured by latitude and longitude. Also, a space shuttle traveling thousands of miles will be far off its target if its heading is off by minutes or even seconds of a degree.

Ex 3 (#48) Perform the calculation.

(a) + (b) 90 °−36 ° 18 ' 47 ' '

______________

Ex 4 (#60) Convert the angle measure to degrees. If applicable, round to the nearest thousandth of a degree. 165 °51 ' 09 ' '

Ex 5 (#72) Convert the angle measure to degrees, minutes, and seconds. Round answer to the nearest second, if applicable. 122.6853 °

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Portions of a degree are measured with minutes and seconds

DefnsOne minute, written 1 ', is 1/60 of a degree. 1'=(1/60)°∨60'=1°One second, written 1 ' ' is 1/60 of a minute. 1' '=(1/60) '=(1/3600)°∨60 ' '=1 '

DefnsAn angle is in standard position if its vertex is at the origin and its initial side is on the x-axis.

Math 335

Ex 6 (#100) Give an expression that generates all angles coterminal with the angle 225 °. Let n represent any integer.

Ex 7Find the angle with least positive measure coterminal with each of the following angles.(a) 260 ° (b) −750 °.

Ex 8 (#124) Locate the point (−2√2 ,2√2 ) and draw a ray from the origin to the point. Indicate with an arrow the angle in standard position having least positive measure. Then find the distance r from the origin to the point, using the distance formula.

Ex 9 (#134) An airplane propeller rotates 1000 times per min. Find the number of degrees that a point on the edge of the propeller will rotate in 1 sec.

Try Problems:

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DefnsAn angle is in standard position if its vertex is at the origin and its initial side is on the x-axis.

Math 335

Sec 1.2: Angle Relationships and Similar TrianglesSketch an angle. How to name it?

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Review Vertical angles (have same measure)

Parallel lines

Transversal

Alternate interior angles (equal)

Alternate exterior angles (equal)

Interior angles on the same side of transversal (suppl.)

Corresponding angles (equal)

Review Types of Triangles Characteristics

Acute TriangleRight Triangle

Obtuse TriangleEquilateral TriangleIsosceles TriangleScalene Triangle

Ex 1 Find the measures of angles 1, 2, 3, and 4 in the figure, given that the lines m and n are parallel.

Ex 2 Find the measure of the marked angles.

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FactThe sum of the measure of the angles of any triangle is 180 °

Conditions for Similar TrianglesFor any triangle ABC to be similar to triangle ¿, the following conditions must hold.

1. Corresponding angles must have the same measure.2. Corresponding sides must be proportional. (That is, the ratios of their corresponding sides must be equal.)

(7 x−5 ) °4

3

21

(9 x+9 ) °

n

m

5 687

1 243

n

m

Math 335

a = (11x + 12)°, b = (4x + 123)°

Ex 3 Find side x with the given information.

a = 13b = 12c = 5d = 26e = 24

Ex 4 A tree casts a shadow 42 m long. At the same time, the shadow cast by a 30-centimeter-tall statue is 66 cm long. Find the height of the tree. Include a sketch.

Sec 3.1 (part I): Radian MeasureSo far, we have seen angles measured in degrees. Another unit of measurement for angles is radians.

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Radian measure is a common unit of measure. In more theoretical work in math, radian measure is preferred as it allows us to treat the domain of trig functions as real numbers, rather than angles. It also simplifies theorems such as the derivative of the sine function. If you plan on moving on to Calculus, you must be able to work in radians. It was wide applications in engineering and science.

Ex1 Draw figures to represent θ=1/2 radian, θ=2 radians and θ=2 π radians.

Ex 2 What quadrant is θ=5 radians in?

Ex 3 Convert each to degree or radian measure.a) b) c) d)

108 °−135 ° 11 π12

−7 π6

Common Measures in Radians and DegreesMust know these equivalences

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NoteIf no unit measure is specified, then the angle is understood to be measured in radians.

Converting Between Degrees and Radians360 °=2π

1. Deg → Rad Multiply by π /180 °2. Rad → Deg Multiply by 180 ° /π

Figure 4 on page 97

DefnAn angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the

radius of the circle has a measure of 1 radian. radian think radius













































































































































































































































































































































Math 335

Try problems: Convert each to degree or radian measure.1) 2) 6

3) 4)

Study these common angles before filling out the following chart without looking. Don’t peek!

Warm Up Name:____________________________Date :______________

Show complete solutions and circle all final answers.

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Degrees Radians (exact)

Radians (approx.)

0 ° 0

30 ° π6

45 ° π4

60 ° π3

90 ° π2

180 ° π

270 ° 3 π2

Degrees Radians (exact)

0 °

30 °

45 °

60 °

90 °

180 °

270 °

Math 335

1. Find the complement to an angle measuring .

2. Find the angle of least positive measure that is coterminal with

3. Locate the point on a rectangular coordinate system.

a) Draw a ray from the origin through the given point.

b) Indicate with an arrow the least positive angle in standard position.

c) Find the distance r from the origin to the point.

4. Find the measure of each angle. (see diagram)

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5. The triangles are similar. Find ∠B. Note that and .

6. Joey wants to know the height of a tree in a park near his home. The tree casts a 38-ft shadow at the same time that Joey, who is 63 inches tall, cast a 42-in shadow. Find the height of the tree.

Sec 1.3: Trigonometric FunctionsWe will define the six trig functions: sine, cosine, tangent, cosecant, secant, and cotangent.

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Let θϵ R define an angle in standard position. Choose any point P ( x , y ) on the terminal side of θ.

We define as follows:

Ex 1 The terminal side of an angle θ in standard position passes through the point (8 ,−6 ). Find the values of the six trig functions of angle θ.

Ex 2 Find the six trig function values of the angle θ in standard position, if the terminal side of θ is defined by 3 x−2 y=0 , x≤ 0.

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Six Trig Functions

sinθ= yr

cosθ= xr

tanθ= yx

, x ≠ 0

cscθ= ry

, y ≠ 0 secθ= rx

, x ≠ 0 cotθ= xy

, y≠ 0

Where r=√x2+ y2

SOH-CAH-TOA

Math 335

Ex 3 Graph y=23

x . Use the graph to determine what tangent really means.

Quadrantal AnglesFormed when terminal side of an angle in standard position lies along one of the axes. Occurs when

θ∈{0+2 kπ , π2

+2 kπ , π+2 kπ , 3 π2

+2kπ }.

Trig Function Values at Quadrantal Angles

θ in degrees θ in radians sin θ cosθ tanθ cot θ secθ csc θ

Ex 4 If is in quadrant III, what are the signs of the following?

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(a) (b) (a)

Ex 5 Find each of the following:

(a) (b) (c)

Ex 6 Evaluate. (a) (b) cos2(−180 °¿)+sin2(−180°¿)¿¿

(c)3sin2 (−90 ° )−5 tan π+6 sec23 π

Note: (Pyth Id)

TRY Evaluate.

(a) (b)

TRY Decide whether each expressions is equal to 0, 1, or -1 or is undefined.(#94) (#96) (#100) (#102)

sin [ n ∙ π ] tan [ (2 n+1 ) ∙ π2 ]cos [n ∙ 2 π ] csc [ n∙ π ]

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Sec 1.4: Using the Definitions of the Trig FunctionsPage 30 of 186

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Note that

csc θ= ry= 1

ry

= 1sin θ

Ex 1 Find each function value.a) b) c) (#10)

tanθ , giventhat cot θ=4 secθ , giventhat cosθ= −2√20

sinθ , giventhat csc θ=√243

Signs of Function Values

ALL STUDENTS TAKE CALCULUS

Ex 2 Determine the signs of the trig functions of an angle in standard position with the given measure.a) b) c) d)

84 °−15° 6π7

8 π7

Ex 3 Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions.a) b) c) (#36)tanθ>0 , cscθ<0sin θ>0 , csc θ>0 cosθ<0 , sinθ<0

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Reciprocal Identities

sin θ= 1cscθ

cosθ= 1secθ

tanθ= 1cotθ

csc θ= 1sin θ

sec θ= 1cosθ

cot θ= 1tanθ

Math 335

Find the Ranges of Trig Functions

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Ranges of Trig Functions

Ex 4 Decide whether each statement is possible.a) b) c) d)

cot θ=−0.999 cosθ=−1.7 csc θ=0sin θ=−π8

Ex 5 Suppose that θ is in QIII and tanθ= 85 . Find the values of the other 5 trig functions.

Derive Pythagorean Identities

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Pythagorean IdentitiesFor all angles θ for which the function values are defined, the following identities hold.

sin2 θ+cos2θ=1 tan2 θ+1=sec2θ 1+cot 2θ=csc2θ

Math 335

Derive Quotient Identities

Ex 6 Find cosθ and tanθ, given that sin θ=−√23

and cosθ>0. Check your answer. Use identity.

Ex 7 Find sin θ and cosθ, given that cot θ=−724 and θ is in QII.

Ex 8 (#74) Find the remaining 5 trig function values if csc θ=2 and θ is in QII.

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Quotient IdentitiesFor all angles θ for which the denominators are not zero, the following identities hold.

sin θcosθ

=tanθ cosθsin θ

=cot θ

Math 335

Sec 2.1: Trigonometric Functions of Acute Angles

Another way to define trigonometric functions based on the ratios of the lengths of the sides of a right triangle.

Ex 1 Find the sine, cosine, and tangent values for angles A and B in the figure.

Note that sin A=cos B and cos A=sin B. Always true for acute angles in a right triangle.

Since angles A and B are complementary and sin A=cos B, sine and cosine are cofunctions.

Ex 2 Write each function in terms of its cofunction.a) sin 9 ° b) cot 76 ° c) csc 45 °

Ex 3 Find one solution for each equation. Assume all angles involved are acute angles. a) cot (θ−8 ° )=tan (4 θ+13 °¿)¿ b) sec (5 θ+14 ° )=csc (2θ−8 ° )

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Right-Triangle-Based Definitions of Trig Functions

sin A= yr= side opposite of A

hypotenusecsc A= r

y= hypotenuse

side opposite of A

cos A= yx

=side adjacent ¿ A ¿hypotenuse

sec A= rx= hypotenuse

side adjacent ¿A ¿

tan A= yx

= side opposite of Aside adjacent ¿

A ¿cot A= xy=side adjacent ¿ A ¿

side opposite of A

SOH-CAH-TOA

A x

yr

85

C

B

A36

77

Cofunction IdentitiesFor any acute angle A, cofunction values of complementary angles are equal.

sin A=cos (90 °− A ) sec A=csc (90 °− A ) tan A=cot (90 °−A )cos A=sin (90 °− A )csc A=sec (90 °− A )cot A=tan (90 °−A )

Math 335

Ex 4 Determine whether each statement is true or false.

a) tan25 °< tan23 ° b) csc 44 °<csc 40 °

Special Angles

STUDY TRIG VALUES WELL! CH 6 (INVERSE STUFF) WILL BE IMPOSSIBLE TO DO WITHOUT BEING PROFICIENT WITH IT. NEEDS TO BE NEARLY SECOND NATURE!!

Function Values of Special Angles

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Degrees Radians sin θ cosθ tanθ csc θ secθ cot θ0 ° 0

30 ° π6

45 ° π4

60 ° π3

90 ° π2

180 ° π

270 ° 3π2

Math 335

Ex 5 Find the six trig function values for a 30 ° angle.

Ex 6 Evaluate. (Note: These problems are in degrees but almost everything in calculus is done in radians.)a) b) c) d) e)

sin 145 ° cos (−60° ) cos150 ° tan 420 ° sin(−150°¿)¿

f) g) h) i) j)

tan (−45 ° ) csc (−60° ) sec120 ° cot (−30 ° )csc 45 °

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Sec 2.2: Trigonometric Functions of Non-Acute Angles

Ex 1 Find the reference angle for each angle.a) 190 ° b) 883 ° c) 11 π /6

Ex 2 Find the exact values of the 6 trig functions for 135 °.

Sec. 3.1 (part II) Ex 3 Evaluate.

a) sin(−5 π6 ) b) cos 5 π

6 c) cot 13 π3

Ex 4 Evaluate sin2 45 °+3 cos2135 °−2 tan 225 °.

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Defn A reference angle for an angle θ, written θ ' , is the positive, acute angle made by the terminal side of angle θ and the x-axis.

Math 335

Ex 5 Evaluate each function by first expressing the function in terms of an angle between 0 ° and 360 °.a) sin 585 ° b) cot (−930 ° )

Ex 6 Find all values of θ, if θ is in the interval [ 0 ° ,360 ° ) and sin θ=−√32

.

TRY Find the exact values of the 6 trig functions for each angle. Always start with a sketch!

(a)

(a)

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Sec 3.2: Applications of Radian Measure

From geometry, we know the arc lengths are proportional to the measure of their central angles.

NOTE: In degrees, the arc length formula is s= θ360

(2πr )= θπr180 .

Ex 1 A circle has radius 25.60 cm. Find the length of the arc intercepted by a central angle having each of the following measures.a) b)

7 π8

radians54 °

Ex 2 Eerie, Pennsylvania is approximately due north of Columbia, South Carolina. The latitude of Eerie is 42 ° N , while that of Columbia is 34 ° N . Find the north-south distance between the two cities. (The radius of Earth is 6400 km.)

*Lattitude gives the measure of a central angle with vertex at Earth’s center, whose initial side goes through the equator and whose terminal side goes through the given location.

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Arc LengthThe length s of the arc intercepted on a circle of radius r by a central angle of measure θ radians is given by the product of the radius and the radian measure of the angle, or

s=rθ , θ∈radians

Math 335

Ex3 A rope is being wound around a drum with radius 0.327 m. How much rope will be wound around the drum if the drum is rotated through an angle of 132.6 °?

Ex 4 Two gears are adjusted so that the smaller gear drives the larger one, as shown in the figure. If the smaller gear rotates through angle of150 °, through how many degrees will the larger gear rotate?

Proof of Area of a sector:

Ex 5 Find the area of a sector of a circle having radius 15.20 ft and central angle 108.0 °.

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5.4 in3.6 in

Area of a SectorThe area A of a sector of a circle of radius r and central angle θ is given by the following formula.

A=12

r2θ ,θ is∈radians

Math 335

Sec 3.3: The Unit Circle and Circular FunctionsIn section 1.3, we defined the 6 trig functions where the domain was a set of angles in standard position. In advanced math courses, it is necessary that the domain consist of real numbers.

The unit circle is symmetric with respect to the x-axis, y-axis, and origin. We can use symmetry and trig function values in the first quadrant to evaluate trig functions in the other 3 quadrants.

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Circular FunctionsFor any real number s represented by a directed arc on the unit circle,

sin s= ycos s=x tan s= yx

, x≠ 0

csc s= 1y

, y ≠ 0 sec s=1x

, x≠ 0cot s= xy

, y≠ 0

Domain of Circular FunctionsSine and Cosine: (−∞, ∞ )

Tangent and Secant: {s|s≠ (2n+1 ) π2

, n∈Z }Cotangent and Cosecant: {s|s≠ nπ , n∈Z }

Math 335

Ex 1 Evaluate.a) b) c) d)

sin 4 π3

cos 4 π3

tan(−9 π4 )sin 11π

6

e) f) g) h)

sec 23 π6

csc 13 π3

cot(−5 π6 )sec ( π

2 )

Note: For calculator problems, be sure the setting is in radian mode.

Ex 2

a) Find the approximate value of s∈[0 , π2 ] if sin s=0.3210.

b) Find the exact value of s∈[3 π2

, 2 π ] if tan s=−√33

.

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Sec 3.4: Linear and Angular SpeedSuppose a point is moving along in a circular path. It has both linear speed and angular speed.

rate= distancetime

Ex 1 Suppose that P is on a circle with radius 15 in. and ray OP is rotating with angular speed π12

radians per second.a) Find the angle generated by P in 10 sec.

b) Find the distance traveled by P along the circle in 10 sec.

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Defns Let P be a point on a circle of radius r moving at a constant speed. The measure of how fast P is changing is the linear speed, v.

v= st

, where s is thelength of the arc traced by P at time t

Let θ be the angle formed by an angle in standard position whose terminal side contains P. The measure of how fast θ changes is its angular speed. Angular speed, denoted ω, is given as

ω= θt

, where θ is∈radians

Note:

v= st=rθ

t=rω

Math 335

c) Find the linear speed of P in inches per second.

Ex 2 A belt runs a pulley of radius 5 in. at 120 revolutions per minute. a) Find the angular speed of the pulley in radians per second.

b) Find the linear speed of the belt in inches per second.

Ex 3 A satellite traveling in a circular orbit approximately 1800 km above the surface of Earth takes 2.5 hrs to make an orbit. (The Earth’s radius is 6400 km.)

a) Approximate the linear speed of the satellite in kilometers per hour. A sketch will help!

b) Approximate the distance the satellite travels in 3.5 hrs.

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Sec 2.4: Solving Right TrianglesMost values obtained for trig applications are not exact. We will round our final answers using significant digits. That is, we round the final answer to the same number of significant digits as the number with the least number of significant digits.In the following numbers, the sig figs are identified in color/bold.

408 21.5 17.00 6.700 0.00 250.0 984074 00

Ex 1 (#12) Solve the right triangle. That is, find the measures of all angles and sides of the triangle. Provide angles answers in degrees and minutes. (Round using sig figs.)

Angles of Elevation or DepressionIn applications of right triangles, the angle of elevation from point X to Y (above X ) is the acute angle formed by XY and a horizontal ray with endpoint X . The angle of depression from point X to Y (below X ) is the acute angle formed by XY and a horizontal ray with endpoint X .

Both angles are measured between the line of sight and a horizontal line (‘x-axis’).

Ex 2 (#56) The length of the shadow of a flagpole 55.20 ft tall is 27.65 ft. Find the angle of elevation of the sun to the nearest hundredth of a degree.

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B

CA b31 ° 40 '

35.9 kmc

Y

XHorizontal

Angle of Elevation

Y

XHorizontal

Angle of Depression

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Section 2.4Name: _________________________________

Date :_____________________

Practice Word Problems on Right Triangles

Solve the following problems. You need to attach a sketch for each problem and label clearly.

1) A guy wire is attached to a 100-ft tower that is perpendicular to the ground. The wire makes an angle of 55 with the ground. What is the length of the wire?

2) A fire is sighted from a fire tower in Wayne National Forest. The ranger found that the angle of depression to the fire is 22. If the tower is 75 meters tall, how far is the fire from the base of the tower?

3) A surveyor is 100 meters from a building. He finds that the angle of elevation to the top of the building is 23. If the surveyor’s eye level is 1.55 meters above the ground, find the height of the building.

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4) From the top of the lighthouse, the angle of depression to a buoy is 25. If the top of the lighthouse is 150 feet above sea level, find the distance from the buoy to the foot of the lighthouse.

5) A ship travels 50 km on a bearing of 27, the travels on a bearing of 117 for 140 km. Find the distance traveled from the starting point to the ending point.

Sec 2.5: Further Applications of Right TrianglesPage 50 of 186

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For full credit on quizzes/exams an accurate, correctly labeled sketch must be drawn. Use your sketch to check reasonability of your answer.

Bearing is an important concept used in navigation and is measured in a clockwise direction from due north.

Sample (true) bearings: 33 ° ,164 ° ,229° , 304 °

We may also express bearings by using a north-south line and using an acute angle to show the direction, either east or west, from the line.

Sample (conventional) bearings: N 42° E , S31° E , S 40 °W , N 52° W

Ex 1 (#12) An observer of a radar station is located at the origin. Find the bearing of an airplane located at (2 , 2 ). Express the bearing using both methods.

Ex 2 (#18) Two ships leave a port at the same time. The first ship sails on a bearing of 52 ° at 17 knots and the second on a bearing of 322 ° at 22 knots. How far apart are they after 2.5 hr?

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Ex 3 (#34) Debbie Glockner-Ferrari, a whale researcher, is watching a whale approach directly toward a lighthouse. When she first begins watching the whale, the angle of depression to the whale is 15 °50 ' . Just as the whale turns away from the lighthouse, the angle of depression is 35 ° 40 '. If the height of the lighthouse is 68.7 m, find the distance traveled by the whale as it approached the lighthouse.

Sec 7.1: Oblique Triangles and the Law of Sines

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In section 2.4, we solved right triangles. We now extend the concept to all triangles.

Congruence Axioms

Whenever SAS, ASA, or SSS is given, the triangle is unique.

An oblique triangle is a triangle that is not a right triangle. Information sufficient to solve an oblique triangle: (1) one side (2) any other two measures

Data Required for Solving Oblique Triangles

Derive the Law of Sines

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If three sides of one triangle are equal, respectively, to three sides of a second triangle, then the triangles are congruent.Side-Side-Side

SSS

If two angles and the included side of one triangle are equal, respectively, to two angles and the included side of a second triangle, then the triangles are congruent.

Angle-Side-AngleASA

If two sides and the included angle of one triangle are equal, respectively, to two sides and the included angle of a second triangle, then the triangles are congruent.

Side-Angle-SideSAS

Use Law CosinesSec 7.3

Use Law of SinesSec 7.1 & 7.2

Case 4 Three SidesSSS

Case 3Two Sides

One Angle included between the two sidesSAS

Case 2Two Sides

One Angle not included between the two sidesThis may lead to more than one triangle

SSA

Case 1One Side

Two AnglesSAA or ASA

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We can use the same method used to derive the Law of Sines to derive formulas for the area of a rectangle (with an unknown height).

Ex 1 (#4) Find the length of side a.

Ex 2 (#12) Solve the triangle.B=38 ° 40' , a=19.7 cm,C=91 ° 40'

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Law of SinesIn any triangle ABC, with sides a, b, and c,

asin A

= bsin B

= csin C

Note: The ratio above is the diameter of the circumscribed circle of the triangle. See Exercise 53.

Alternative form:

sin Aa

= sin Bb

= sin Cc

Area of a Triangle (SAS)In any triangle ABC, the area A is given by the following formulas.

A=12

bcsin A A= 12

ab sinC A= 12

ac sin B

Math 335

Ex 3 (#26) To determine the distance RS across a deep canyon, Rhonda lays off a distance TR=582 yd . She then finds that T=32 °50 ' and R=102° 20 ' . Find RS.

Ex 4 (#30) Standing on one bank of a river flowing north, Mark notices a tree on the opposite bank at a bearing of 115.45°. Lisa is on the same bank as Mark, but 428.3 m away. She notices that the bearing of the tree is 45.47 °. The two banks are parallel. What is the distance across the river?

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Ex 5. The angle of depression from the top of a building to a point on the ground is . How far is the point on the ground from the top of the building if the building is 252 m high?

Ex 6 Two docks are located on an east-west ling 2,587 feet apart. From dock A, the bearing of a coral reef is . From dock B, the bearing of the coral reef is . Find the distance from dock A to the coral reef.

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Ex 7 (#34) Three atoms with atomic radii of 2.0, 3.0, and 4.5 are arranged as in the figure. Find the distance between the centers of atoms A and C.

Ex 8 (#40) Find the area of the triangle using the formula A=12

bh, and then

verify that the formula A=12

ab sin C gives the same result.

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Ex 9 (#55) Several of the exercises on right triangle applications involve a figure similar to the one shown Use the law of sines to obtain x in terms of α ,β , and d .

Sec 7.3: The Law of Cosines

Ex 1 A triangle has base of length 13 and the other two sides are equal in length. If the lengths are integers, what is the shortest possible length of the unknown side?

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Triangle Side Length RestrictionIn any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. (Sum of two shortest sides must be greater than the longest side.)

Math 335

Derive the Law of Cosines (using the distance formula)

See “Suggested Procedure for Solving Oblique Triangles” on page 310.

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Law of CosinesIn any triangle ABC with sides a ,b , and c, the following hold.

a2=b2+c2−2 bc cos Ab2=a2+c2−2ac cos Bc2=a2+b2−2 abcos C

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Ex 2 Determine whether SAA, ASA, SSA, SAS, or SSS is given. What law would you use?#1 #2 #3 #4

a , b ,∧C A ,C ,∧ca , b ,∧A a , B ,∧C

#5 #6 #7 #8A ,B ,∧c a , c ,∧A a , b ,∧cb , c ,∧A

Ex 3 (#12) Find the measure of θ. (No calculator.)

Ex 4 Solve each triangle.(#16) (#34)

B=168.2 °a=15.1 cmc=19.2 cm

Ex 5 (#40) Points X and Y are on opposite sides of a ravine. From a point third point Z, the angle between the lines of sight to X and Y is 37.7 °. If XZ is 153 m long and YZ is 103 m long, find XY .

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√3

1 1

θ

C

B

8

10 4

A

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Ex 6 (#44) An airplane flies 180 mi from point X at a bearing of 125 °, and then turns and flies at a bearing of 230 ° for 100 mi. How far is the plane from point X .

Ex 7 (#51) A weight is supported by cables attached to both ends of a balance beam, as shown in the figure. What angles are formed between the beams and cables?

Ex 8 (#57) Surveyors are often confronted with obstacles, such as trees, when measuring the boundary of a lot. One technique used to obtain an accurate measurement is the so-called triangulation method. In this technique, a triangle is constructed around the obstacle and one angle and two sides of the triangle are measured. Use this technique to find the length of the property line (the straight line between the two markers) in the figure.

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Ex 9 (#66) Find the area of the triangle ABC. a=22∈. ,b=45∈. , c=31∈.

Sec 7.2: The Ambiguous Case of the Law of SinesRecap: To solve a triangle, we need to to find ____________________________________________.

To solve a triangle, we need to be given AT LEAST _________________________________________.

Six possible cases of 3 given measurements:

I. Angle-Angle-Angle

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Heron’s Formula (SSS)If a triangle has side lengths a ,b , and c with semiperimeter

s= 12

( a+b+c ) ,

then the area A of the triangle is given by the formula:

A=√s ( s−a ) (s−b ) (s−c )

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II. One side and any two angles

III. Two sides and an included angle

IV. Side-Side-Side

V. Two sides and a non-included angle

Why is SSA ambiguous?

Let θ and a be fixed and assume θ is acute.

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a

θ

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Ex 1 (#8) Determine the number of triangles possible with the given parts.B=54 ° , c=28 , b=23

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Ex 2 Find the unknown angles in triangle ABC for each triangle that exists.(#18)

C=82.2 ° ,a=10.9 km , c=7.62 km

Ex 3 Solve the triangle that exists.(#22)

C=52.3 ° , a=32.5 yd , c=59.8 yd

Ex 4(#30) A=51.20° , c=7986 cm ,a=7208 cm

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Ex 5 (#36) The angle of elevation from the top of a building 45.0 ft high to the top of a nearby antenna tower is 15 °20 ' . From the base of the building, the angle of elevation of the tower is 29 °30 ' . Find the height of the tower.

Ex 6 (#38) A pilot flies her plane on a heading of 35 ° 00 ' from point X to point Y which is 400 mi from X. Then she turns and flies on a heading of 145 °00 ' to point Z, which is 400 mi from her starting point X. What is the heading of Z from X , and what is the distance YZ?

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Ex 7 (#40) Use the law of sines to prove the statement is true for any triangle ABC, with corresponding sides a ,b , and c.

a−ba+b

= sin A−sin Bsin A+sin B

Summary of Solving a Triangle

Oblique Triangle What do I do?

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Sec 4.1: Graphs of the Sine and Cosine Functions

Graph of y=sin x

x y=sin x ( x , y )0 0 (0 , 0 )

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π6

12 ( π

6, 12 )

π2 1 ( π

2, 1)

5 π6

12 ( 5 π

6, 12 )

π 0 ( π , 0 )7 π6

−12 ( 7π

6,− 1

2 )3π2 −1 ( 3π

2,−1)

11π6

−12 ( 11π

6,−1

2 )2 π 0 (2π ,0 )

Ex 1 Graph each function.

y=3sin x

y=3sin x+1

y=sin (2 x )

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DefnA periodic function is a function f such that f ( x )=f ( x+np ) for every real number x in the domain of f , every integer n, and some positive real number p. The least positive value of p is the period of the function.Periodic functions “repeat” with a regular pattern.

Math 335

Ex 2 Graph each function.

y=cos x

y=−cos x+1

y=2cos (3 x )

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Properties of the Sine Function

1) dom sin x=¿

2) rng sin x=¿

3) The sine function is an ______ function, as the symmetry of the graph with respect to the origin

indicates. Moreover, sin (−x )=¿________ ∀ x∈domsin x.

4) The sine function is periodic, with period ______.

5) The x-intercepts occur at x=0 , ± π , ± 2 π , ±3 π , …

6) The y-intercept is (0,0 ).

7) The maximum value is 1 and occurs at x=…−3π2

, π2

, 5 π2

, 9π2

, …

8) The minimum value is -1 and occurs at x=…−π2

, 3 π2

, 7 π2

, 11 π2

,…

Properties of the Cosine Function

1) dom cos x=¿

2) rngcos x=¿

3) The cosine function is an ______ function, as the symmetry of the graph with respect to the y-axis

indicates. Moreover, cos (−x )=¿________ ∀ x∈domcos x.

4) The cosine function is periodic, with period ______.

Math 335

Calculate the amplitude of y=3sin x and y=2cos x. Provide a sketch of each graph.

Ex 3 Make a conjecture about the graphs of y=sin x and y=cos x.

Conjecture:

sin x=¿

Note: Because of this relationship, the graphs of y=A sin ( ωx ) or y=A cos (ωx ) are referred to as sinusoidal graphs.

Ex 4 Find the period of y=sin x , y=cos x , y=−sin (2 x ) , and y=2cos (3 x ).

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Properties of the Cosine Function

1) dom cos x=¿

2) rngcos x=¿

3) The cosine function is an ______ function, as the symmetry of the graph with respect to the y-axis

indicates. Moreover, cos (−x )=¿________ ∀ x∈domcos x.

4) The cosine function is periodic, with period ______.

DefnThe amplitude of a periodic function is half the difference between the maximum and minimum values. The graph of y=a sin x or y=a cos x, with a≠ 0, will have the same shape as the graph of y=sin x or y=cos x, respectively, except with range [−|a|,|a|]. The amplitude is ______.

Period

For b>0, the graph of y=sin bx will resemble that of y=sin x , but with period 2 πb . Similarly for

y=cosbx and y=cos x.

Math 335

Ex 5 Graph each function over a two-period interval. Give the period and amplitude.

(#38’)

y=23

sin π4

x f ( x )=−2 cos( 13

x)

Ex 6 Write the function that describes each graph. Assume no phase (horizontal) shift.

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(a) (b)

(c)(d)

(e)

(f)

(g)

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Sec 4.2: Translations of the Graphs of the Sine and Cosine Functions

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Recall From Algebra:

Combinations of Translations

y=a sin [b ( x−d ) ]+c∨ y=acos [b ( x−d ) ]+c , where b>0

Ex 1 Find the amplitude, the period, any vertical translation, and any phase shift of the graph of each function.

(a) (b)

(#26) (#30)

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DefnWith circular functions, a horizontal translation is called a phase shift. In the function f ( x ), the input value is called the argument.

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(c ) y=3 sin(x+ π2 )(d ) f ( x )=−cos [π ( x−1

3 )]

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(#48) (#52)

(e ) f ( x )=1− 23

sin 34

x ( f ) y=1+ 23

cos 12

x

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Ex 2 Match the function with its graph.

A ¿ y=sin (x−π2 )B ¿ y=sin (x+ π

2 )C ¿ y=cos(x− π2 ) D¿ y=cos( x+ π

2 )E ¿ y=sin x+2F ¿ y=sin x−2G ¿ y=cos x+2 H ¿ y=cos x−2

(a) (b)

_____________________________________________

(c) (d)

_________________________ ____________________

(e) (f)

_________________________ ____________________

(g) (h)

_________________________ ____________________

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Sec 4.3: Graphs of the Tangent and Cotangent FunctionsBecause the tangent function has period π, we only need to determine the graph over some interval of length π. (The rest of the graph will consist of repetitions over some interval of length π.) Because the

tangent function is not defined at …,−3 π2

,− π2

, π2

, 3 π2

,…, we will concentrate on the interval

(−π2

, π2 ).

x y=tan x ( x , y )−π3 −√3≈−1.73 (−π

3,−√3)

−π4 −1 (−π

4,−1)

−π6

−√33

≈−0.58 (−π6

,−√33 )

0 0 (0 , 0 )π6

√33

≈ 0.58 ( π6

, √33 )

π4 1 ( π

4, 1)

π3 √3≈ 1.73 ( π

3,√3)

What happens when x-values approach π /2?

That is, what is lim

x→ π2

−¿

tan x ? Seetable¿

¿

x sin x cos x y=tan xπ3

≈ 1.05 √32

12 √3≈ 1.73

1.5 0.9975 0.0707 14.11.57 0.9999 0.000796 1255.8

1.5707 0.9999 0.000096 10,381π2

≈ 1.5708 1 0 Undefined

The tangent function has a vertical asymptote everywhere it’s undefined, that is, when cos x=0. On

[−π2

, π2 ], it has a vertical asymptote at _____________________________.

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Properties of the Tangent Function

1) dom tan x=¿

2) rng tan x=¿

3) The tangent function is an ______ function, as the symmetry of the graph with respect to the origin

indicates. Moreover, tan (−x )=¿________ ∀ x∈dom tan x.

4) The tangent function is periodic, with period ______.

5) The x-intercepts occur at x=0 , ± π ,± 2π , ±3 π , …

Math 335

Ex 1 Graph each.f ( x )=2 tan x y=−3 tan (2x )

We define the cotangent function as we did the tangent function. The period of y=cot x is also π and because cotangent is not defined for integer multiples of π, we will concentrate on the interval (0 , π ).

x y=cot xπ6 √3 ≈ 1.73

π4 1

π3

√33

≈ 0.58

π2 0

2 π3

−√33

≈−0.58

3 π4 −1

5 π6 −√3≈−1.73

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Properties of the Tangent Function

1) dom tan x=¿

2) rng tan x=¿

3) The tangent function is an ______ function, as the symmetry of the graph with respect to the origin

indicates. Moreover, tan (−x )=¿________ ∀ x∈dom tan x.

4) The tangent function is periodic, with period ______.

5) The x-intercepts occur at x=0 , ± π ,± 2π , ±3 π , …

Math 335

limx→ 0+¿ cot x lim

x→ π−¿ cot x¿¿¿

¿

The cotangent function has a vertical asymptote everywhere it’s undefined, that is, when sin x=0. On [ 0 , π ], it has a vertical asymptote at _____________________________.

Ex 2 Graph each.

f ( x )=1−12

tan (2 x ) f ( x )=2cotx−1 f (x )=1−cot( 12

x)

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Properties of the Cotangent Function

1) dom cot x=¿

2) rngcot x=¿

3) The cotangent function is an ______ function, as the symmetry of the graph with respect to the

origin indicates. Moreover, cot (−x )=¿________ ∀ x∈dom f .

4) The cotangent function is periodic, with period ______.

5) The x-intercepts occur at x=± π2

,± 3 π2

,± 5π2

, …

6) The y-intercept is ________.

7) Vertical asymptotes occur at x=0 , ± π , ± 2 π , ±3 π , …

Math 335

Ex 3 (#43’) If c is any number, how many solutions does the equaton c= tan x have in the interval [−2 π ,2π ]? [−2 π , 2 π )?

Ex 4 Write the function for each graph. Assume no phase shift. (a) (b)

_________________________ _______________________

(c) (d)

______________________________________________________

(e)

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________________________________

Sec 4.4: Graphs of the Secant and Cosecant FunctionsThe secant and cosecant functions, sometimes referred to as reciprocal functions, are graphed by making use of the reciprocal identities.

csc x= 1sin x

sec x= 1cos x

Note that secant is an even function. Thus, sec (−x )=sec x ∀ x∈dom sec x.

x y=sec x0 1

± π6

2√33

≈1.2

± π4

√2≈1.4

± π3

2

± 2 π3

−2

± 3 π4

−√2≈−1.4

± 5 π6

−2√33

≈−1.2

± π −1

x y=csc x x y=csc xπ6

2 −π6

−2

π4

√2≈1.4 −π4

−√2 ≈−1.4

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π3

2√33

≈1.2−π3

−2√33

≈−1.2

π2

1 −π2

−1

2 π3

2√33

≈1.2−2 π

3−2√3

3≈−1.2

3 π4

√2≈1.4 −3π4

−√2 ≈−1.4

5 π6

2 −5 π6

−2

Ex 1 Graph each function over a one period interval.

(a ) y=−2csc(x− π2 )(b) y=3 sec (2 x−π )+2

Ex 2 Write the function for each graph.

(a) (b) (c)

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Sec 4.5: Harmonic MotionSimple harmonic motion is an important concept for those studying engineering or physics.

Idea: A weight is attached to a coiled spring. It is pulled down a distance of 20 cm from its equilibrium position and released. The time for one complete oscillation is 4 seconds.

VISUAL:

Ex 1 (#4) A weight on a spring has initial position s (0 )=−4 and period P=1.2 sec. (a) Find a function s given by s ( t )=acos ωt that models the displacement of the weight.

(b) Evaluate s (1 ). Is the weight moving upward, downward, or neither when t=1? Support your results graphically or numerically.

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Simple Harmonic MotionThe position of a point oscillating about an equilibrium position at time t is modeled by either s ( t )=a cosωt or s ( t )=a sin ωt , where a and ω are constants, with ω>0. The amplitude of the motion is |a|, the period is 2 π /ω, and the frequency is ω /2π oscillations per minute. (The number of cycles per unit of time is called the frequency and is equal to the reciprocal of the period.)

Math 335

Ex 2 (#6) A note on the piano has frequency F=110. Suppose the maximum displacement at the center of the piano wire is given by s (0 )=0.11. Find constants a and ω so that the equation s (t )=acos ωt models this displacement. Graph s in the viewing window [ 0 ,0.05 ] by [−0.3 ,0.3 ].

Ex 3 (#10) An object is attached to a coiled spring as in the figure given. It is pulled down a distance of 6 units from its equilibrium position and then released. The time for one complete oscillation is 4 sec.(a) Give an equation that models the position of the object at time t .

(b) Determine the position at t=1.25 sec.

(c) Find the frequency.

Ex 4 (#14) The period in seconds of a pendulum of length L in feet is given by:

P=2 π √ L32

.

How long should the pendulum be to have a period of 1 sec?

Ex 5 (#18) The position of a weight is attached to a spring is s (t )=−4 cos10 t inches after t seconds. a) What is the maximum height that the weight rises above the equilibrium position?

b) What are the frequency and period?

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c) When does the weight first reach its maximum height?

d) Calculate and interpret s (1.466 ).Chapter 4 Trig Graphing Review

Name :______________________________Date :___________________

1. Determine the period of each function.

(a) (b)

(c) (d)

2. Determine the amplitude, phase shift, and range for each function.

(a) (b)

(c) (d)

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3. Graph each of the following trigonometric functions. Graph at least one period. Clearly label the intervals, any asymptotes, etc.

(a) (b)

(c) (d)

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(e) (f)

(g) (h)

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4. Write the function that describes each graph. Assume no phase (horizontal) shift.

(a) (b)

(c)

(d) (e) (f)

(g) (h)

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5. Write the function for each graph. Assume no phase shift.

(a)

(a) (b) (c)

6. Match the function with its graph.

A ¿ y=sin (x−π2 )B ¿ y=sin (x+ π

2 )C ¿ y=cos(x− π2 ) D¿ y=cos( x+ π

2 )E ¿ y=sin x+2F ¿ y=sin x−2G ¿ y=cos x+2 H ¿ y=cos x−2

(a) (b)

(c)

(d) (e)

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Sec 5.1: Fundamental Identities

An identity is an equation that is satisfied by every value in the domain of its variable. We will be verifying trig identities in section 5.2.

Below are identities that must be memorized perfectly for hw, quizzes, and exams. Note that numerical values can be used to help check whether or not an identity was recalled correctly.

Ex 1 (#16) Find sin θ if secθ=72 and tanθ<0.

Ex 2 (#38) Find the remaining five trig functions if cosθ=−14 and sin θ>0.

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Fundamental IdentitiesReciprocal Identities

cot θ= 1tan θ

secθ= 1cosθ

cscθ= 1sinθ

Quotient Identities

tanθ= sin θcosθ

cotθ= cosθsin θ

Pythagorean Identitiessin2 θ+cos2θ=1 tan2 θ+1=sec2θ 1+cot 2θ=csc2θ

Negative-Angle Identitiessin (−θ )=−sin θ cos (−θ )=cosθ tan(−θ¿)=−tan θ ¿

csc (−θ )=−csc θ sec (−θ )=sec θ cot(−θ¿)=−cot θ ¿

WARNING:

tanθ=−34 does NOT imply

Math 335

Ex 3 Write each expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression and all functions are of θ only.a) (#70)

(1+cot θ )cot θ

b) (#76) c) (#82)

(sin θ−cosθ ) (csc θ+secθ )−sec2 (−θ )+sin2 (−θ )+cos2 (−θ )

Ex 4 (#86) Let csc x=−3. Find all possible values of sin x+cos x

sec x

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Include Factoring reviewSec 5.2: Verifying Trigonometric Identities

One of the skills required in more advanced mathematics, especially calculus, is the ability to use identities to write expressions in alternative forms.

Warning: Techniques used to solve equations, such as adding the same term to each side, and multiplying each side by the same term. To avoid the temptation, one strategy is to work with only ONE side and rewrite it until it matches the other side. Your proofs will be graded on not only correctness but presentation as well.

If both sides of an identity is equally complex, the identity can be verified by working independently on the left side and on the right side, but these steps are considered scratch work. You must rewrite the proof using the fact that each step (of one side) of your scratch work is reversible.

Ex 1 (#12) Perform each indicated operation and simplify the result so that are no quotients.1

sin α−1− 1

sin α+1

Ex 2 Factor each trig expression. (Simplify if possible.)a) (#16) b) (#18)

( tan x+cot x )2−( tan x−cot x )2 4 tan2 β+ tan β−3

c) (#22) d)

sin3 α+cos3 α 3 cot3 x+15cot2 x−12 cot x−60

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Ex 3 Each expression simplifies to a constant, a single function, or a power of a function. Use the fundamental identities to simplify each expression.a) (#32) b) (#34)

1tan2 α

+cot α tan α 1− 1sec2 x

Ex 4 Verify that each trig equation is an identity.a) (#42)

sin2 α+ tan2 α+cos2 α=sec2 α

b) (#48) c) (#66)

( sec α−tan α )2=1−sin α1+sin α

1+sinθ1−sin θ

−1−sin θ1+sinθ

=4 tan θ sec θ

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d) (#70)

(1+sin x+cos x )2=2 (1+sin x ) (1+cos x )

e) (#76)

sin2 x (1+cot x )+cos2 x (1−tan x )+cot2 x=csc2 x

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f) (#78) g)

sin3 θ+cos3θ=(cosθ+sin θ ) (1−cosθ sin θ ) 1−sin θ1+sinθ

=( secθ−tanθ )2

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Sec 5.3: Sum and Difference Identities for CosineDoes cos ( A−B )=cos A−cos B? Why or why not?

Derive the formula for cos ( A−B ). (Sum Identity for Cosine) Use the result to find the Difference Identity for Cosine.

Use a cosine identity to derive a cofunction identity below.

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Cosine of a Sum or Differencecos ( A+B )=cos A cos B−sin A sin Bcos ( A−B )=cos A cosB+sin A sin B

Cofunction Identities

cos ( π2

−θ)=sin θ cot( π2

−θ)=tanθ

sin( π2

−θ)=cosθ sec( π2

−θ)=csc θ

tan( π2−θ)=cot θ csc( π

2−θ)=secθ

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Ex 1 Find each exact value.a) (#12) b) (#16)

cos ( π12 )cos 7 π

9cos 2 π

9−sin 7 π

9sin 2π

9

Ex 2 (#40) Find one angle that satisfies the equation.

cos θ=sin( θ4

+3 °)

Ex 3 (#48) Write the expression as a function of θ. cos ( 90°+θ )

Ex 4 (#56) Find cos ( s+t ) and cos ( s−t ).

cos s=√24

∧sin t=−√56

, s∧t ∈quadrant IV

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Ex 5 (#58) Determine if the statement is true or false.cos (−24 ° )=cos (16 ° )−cos (40 ° )

Ex 6 Verify that each equation is an identity.(#70)

1+cos2 x−cos2 x=cos2 x

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Sec 5.4: Sum and Difference for Sine and TangentDerive the Sum Identity for Sine. Next, use the result to find the Difference Identity for Sine.

Derive the Sum Identity for Tangent.

Ex 1 Find each exact value.(#16) (#18)

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Tangent of a Sum or Difference

tan ( A+B )= tan A+ tan B1−tan A tan B

tan ( A−B )= tan A−tan B1+tan A tan B

Sine of a Sum or Differencesin ( A+B )=sin A cos B+cos A sin Bsin ( A−B )=sin A cos B−cos A sin B

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sin(−5 π12 ) tan (−7 π

12 )

Ex 2 Write each function as a single function of x or θ.(#28) (#36)

cos (θ−30 ° )sin ( 3π4

−x )

Ex 3 (#48) Given:

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cos s=−1517

∧sin t=45

, s∈QII∧t∈QI

Find (a) sin ( s+t )

(b) tan ( s+t )

(c) the quadrant of s+t

Ex 4 Find each exact value.(#56)

tan165 ° sin(−13 π12 )

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Ex 5 Verify each identity.(#62) PP

sin ( x+ y )+sin ( x− y )=2sin xcos y

(#66)

sin (s+t )coss cos t

=tan s+ tan t

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Sec 5.5: Double-Angle IdentitiesDerive the double-angle identity for cosine.

cos (2 A )=¿

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Sum-to-Product Identities

sin A+sin B=2 sin( A+B2 )cos ( A−B

2 )sin A−sin B=2cos( A+B

2 )sin( A−B2 )

cos A+cosB=2cos( A+B2 )cos ( A−B

2 )cos A−cos B=−2sin ( A+B

2 )sin( A−B2 )

Product-to-Sum Identitiescos A cos B=1

2 [cos ( A+B )+cos ( A−B ) ]

sin A sin B=12 [cos ( A−B )−cos ( A+B ) ]

sin A cos B= 12 [sin ( A+B )+sin ( A−B ) ]

cos A sin B= 12 [sin ( A+B )−sin ( A−B ) ]

Double-Angle Identities

cos (2 A )=cos2 A−sin2 A cos2 A=1−2 sin2 Acos (2 A )=2cos2 A−1 sin (2 A )=2 sin A cos A

tan (2 A )= 2 tan A1−tan2 A

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Ex 1 Find the values of the sine and cosine functions for each angle measure.a) (#12) b) (#14)

2θ , given cosθ=√35

∧sin θ>0 θ , givencos2θ=34∧θ terminates∈QIII

Ex 2 Verify each identity.(#30) (#32)

cos2 x=1−tan2 x1+tan 2 x

cot A− tan Acot A+ tan A

=cos2 A

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(#36)

cot θ tan (θ+π )−sin (π−θ ) cos ( π2

−θ)=cos2 θ

Ex 3(#38) (#40)

2 tan 15 °1−tan215 °

1−2sin2 22 12

°

(#46) (#48)18

sin 29.5 ° cos29.5° cos22 x−sin2 2x

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Ex 4 (#50) Express cos3 x as a trig function of x.

Ex 5 (#60) Write as a sum or difference of trig functions. 5 cos3 xcos 2 x

Ex 6 (#66) Write as a product of trig functions. sin 102°−sin 95 °

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Chapter 5 Identity Review

Name:__________________________

Date:_____________

Verify each of the following identities:

1.

2.

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3.

4.

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5.

6.

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7.

8.

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Sec 5.6: Half-Angle IdentitiesDerive the half-angle identity for sine, cosine and tangent.

sin A2

=¿

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Ex 1 (#12) Evaluate. sin 195 °

Ex 2 Find each.a) (#24)

cos x2

, given cot x=−3 , with π2

<x<π

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Half-Angle IdentitiesChoose the correct sign based on the function under consideration and the quadrant of A/2.

cos A2

=±√ 1+cos A2

sin A2

=±√ 1−cos A2

tan A2

=± √ 1−cos A1+cos A

tan A2

=sin A

1+cos Atan A

2=

1−cos Asin A

NOTE: The last two formulas have the advantage of not requiring a sign choice.

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b) (#26)

cot θ2

, giventhat tan θ=−√52

, with90 °<θ<180 °

c) (#30)

sin x , givencos 2 x=23

,with π<x< 3 π2

Ex 3 Write as a single trig function.(#38) (#40) (#44)

sin 158.2 °1+cos158.2°

±√ 1+cos20 α2

±√ 1−cos 3θ5

2

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Ex 4 Verify each identity.(#48) (#52)

sin 2x2sin x

=cos2 x2−sin 2 x

2cos x=

1−tan2 x2

1+tan 2 x2

Ex 5

(#53) Use the half-angle identity to derive the equivalent identity

by multiplying the numerator and the denominator by 1−cos A (the ‘conjugate’).

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tan A2

= sin A1+cos A tan A

2=1−cos A

sin A

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Sec 6.1: Inverse Circular FunctionsTry Problem:

Given . (a) Is this a function? (b) Is it a one-to-one function?

(c) Find its inverse.

Fundamental Topics of Trig: Graphs, identities, inverse functions and notation, solving equations. All require proficiency with trig evaluation.

Review

Ex 1 Find the inverse of f ( x )=x2. What can we conclude?

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DefnsA function is a relation (a set of ordered pairs) where each x-value corresponds to exactly ONE y-value. A one-to-one (1-1) function is a function where each y-value corresponds to exactly ONE x-value.

If f is a function, the inverse of f is a relation consisting of all ordered pairs ( y , x ) where ( x , y )∈ f . If f is 1-1, then its inverse is a function. The inverse function, denoted f−1, is defined as

f−1= {( y , x )|(x , y )∈ f }

Warning!

f−1≠ 1f

Horizontal Line Test (HLT)A function is one-to-one if every horizontal line interests the graph of the function at most once.

4 Biggies of Inverse FunctionsI. ' x ' and ' y ' values switch places

II. inverses reflect in the line y=x III. f ( f −1 ( x ) )=x and f−1 ( f ( x ) )=x IV. The inverse is a function iff f ( x ) is one-to-one.

How to Find the Inverse of a 1-1 FunctionStep 1: Replace f ( x ) with yStep 2: x↔ y (interchange x and y)Step 3: Solve for y.

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Consider f ( x )=sin x. Define an inverse for sine.

Note: The argument of inverse functions are real numbers and their output are angles. In particular, arcsine takes in a real number but ‘spits out’ an angle.

Warning: sin−1 x ≠ (sin x )−1

That is, sin−1 x≠ 1sin x . However, sin2 x=¿ ________=________________

Notation

arcsin 12=θ means find _______ (ONE only) such that ______________________.

vs

sin θ=12

which means_____________________________________. This has _____________ solutions.

Note: _________ is the ONLY correct solution to arcsin 12 .

Why call it arcsin? Because θ (in radians) is the ____________________________________________________.

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Inverse Sine Functiony=sin−1 x or y=arcsin x means that x=sin y , for −π /2≤ y ≤ π /2.

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Analogy to Elementary Algebra:

Cosine is not 1-1 either (on its natural domain). Define an inverse for cosine.

Inverse Tangent:

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Inverse Cotangent, Secant, and Cosecant Functionsy=cot−1 x or y=arccot x means that x=cot y , for −0< y<π .

y=sec−1 x or y=arcsec x means that x=sec y , for −0≤ y≤ π , y ≠ π /2.

y=csc−1 x or y=arccsc x means that x=csc y , for −π /2≤ y ≤ π /2 , y≠ 0.

Inverse Tangent Functiony=tan−1 x or y=arctan x means that x=tan y, for −π /2< y<π /2.

Inverse Cosine Functiony=cos−1 x or y=arccos x means that x=cos y, for −0 ≤ y≤ π.

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Note: The inverse secant and inverse cosecant functions are sometimes defined with different ranges. We use intervals that match those of the inverse cosine and inverse sine functions (except for the missing point).

Ex 2 (#8’) Consider the inverse cosine function, defined by y=cos−1 x, or y=arccos x.

a) Is this function increasing or decreasing?

b) arccos(−12 )=2 π

3 . Why is arccos(−12 ) not equal to

−4 π3 ?

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InverseFunction Domain Range

Interval Quadrants

y=sin−1 x [−1,1 ] [−π2

, π2 ]

y=cos−1 x [−1,1 ] [ 0 , π ]

y=tan−1 x (−∞, ∞ ) (−π2

, π2 )

y=cot−1 x (−∞, ∞ ) (0 , π )

y=sec−1 x (−∞,−1 ]∪ [ 1, ∞ ) [0 , π2 )∪( π

2, π ]

y=csc−1 x (−∞,−1 ]∪ [ 1, ∞ ) [−π2

,0)∪(0 , π2 ]

Math 335

Ex 3 (#11) Is sec−1 a calculated as cos−1 1a or as

1cos−1a

?

Ex 4 Evaluate each if possible.(#14-36 evens)

(a) y=sin−1 (−1 )(b) y=arctan (−1 )

(c ) cos−1(−12 )=¿ (d) arccot (−√3 )=¿

(e) sec−1 (−√2 )=( f )arccsc(−12 )=¿

(g) arcsin (−√32 )=¿ (h) arccsc (−2 )=¿

Ex 5 Evaluate if possible. State answer in degrees.(#38) (#48)

(a) tan−1 √3 (b )cos−1 (−2 )

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Ex 6 Evaluate.

(a )sin (sin−1 √32 ) (b )sin−1(sin π

6 ) (c ) cos(cos−1 √32 )( d )cos−1(cos π

3 )

(#80) (#84)

(e ) sin(arccos 14 )( f ) cos(2sin−1 1

4 )

(#86)

(g) tan(2cos−1 14 ) (h)cos−1(cos (−5 π

3 ))

(#92)

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(i) cos (sin−1 35

+cos−1 513 )

(j)cos−1(sin 7 π6 )=¿

Ex 7 Write as an algebraic (non-trigonometric) expression in u , u>0.(#102)

cot (arcsin u )

(#106)

sec(cos−1 u√u2+5 )

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Sec 6.2: Trigonometric Equations I Ex 1 Solve each equation for all x such that x∈ [ 0,2π ).(#13) (#22)

2sin x+3=4 2cos2 x−cos x=1

Ex 2 Solve each equation for θ such that θ∈ [0,360 ° ). Note: On exam, numbers will work out nice (no calculators)

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Ex 3 Solve each equation (x in radians and θ in degrees) for all exact values. Note: HW says round. Obtain exact answer first (this is your quiz/exam answer) then round (this is your hw answer). Write answers using the least possible nonnegative angle measures. Round degrees to the nearest tenth and radians to four decimal places.

(#44)

(a ) tanθ+1=0(b)

(#50)

(c ) 4cos2 x−1=0(d)

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(e)

Ex 4 Grade the solution.Problem: Solve for all values of x∈ [ 0,2 π ) that satisfy cos2 x=1.

Solution:

cos 2x2

=12⇒cos x=1

2⇒ x= π

3

Ex 5 Solve for all values of x∈ [ 0,2 π ).a)

6cot x cos x−3 cot x=0

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b) c)

tanθ+√3=02 sin2θ−3 sin θ+1=0

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Factoring & Zero Factor Property ReviewName :_______________________

Date :_________________Factor Completely:

1. 24 8x x 2.

3. 2 5 5x p x 4.

5.225 m 6.

7. 2 6 9x x 8.

9. 2 7 12x x 10.

11. 2 4 12x x 12.

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13. 23 2x x 14.

2 26 35 11x xy y

Solve:Remember, if the equations are quadratic in form you have the following 4 tools:

1. Quadratic Formula2. Factoring (ZFP)3. Complete the Square4. Square Root Property – Don’t forget the ___________5.

15. 16.

17. 18

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Sec 6.3: Trigonometric Equations II Ex 1 Solve each equation for x such that xϵ [0,2 π ) and for θ such that θϵ [ 0,360 ° ).(#16) (#18)

tan 4 x=0 cos2 x−cos x=0

(#20) (#24)

sin2 x2−2=0 sin x cos x= 1

4

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Ex 2 Solve each equation (x in radians and θ in degrees) for all exact values. Note: HW says round. Obtain exact answer first (this is your quiz/exam answer) then round (this is your hw answer). Write answers using the least possible nonnegative angle measures.(#34) (#36)

sin 2 x=2 cos2 x cos x=sin 2 x2

(#38’) Solve for only values in [ 0,2 π ) (#42) Solve for only values in [ 0,2 π )

4 cos2 x=8 sin xcos x sin x2+cos x

2=1

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Sec 6.4: Equations Involving Inverse Trigonometric Functions Ex 1 Solve.(#12) (#16)

y=−sin x3

, for x∈[−3 π2

, 3 π2 ] y=tan (2x−1 ) , for x∈(1

2− π

4, 1

2+ π

4 )

(#20) (#28)

y=4+3cos x , x∈ [0 , π ] 4 π+4 tan−1 x=π

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(#30) (#34)

arccos(x−π3 )= π

6cot−1 x=tan−1 4

3

(#38) (#40)

arccos x+2arcsin √32

=π3

arcsin 2 x+arcsin x= π2

(#42) PP

sin−1 x+tan−1 x=0

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Ans: {0 }

Solving Trigonometric Equations

I. Solve each equation for if 0 °≤θ<360° .

1) 2 cosθ=1 2) 2 cosθ−√3=0

3) 2 tan θ+2=0 4) √3 cot θ−1=0

II. Solve each equation for if 0≤θ<2π .

5) 4 sin t−√3=2sin t 6) 5 cos t +√12=cos t

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7) (sin x−1)(2 sin x−1)=0 8) 2 sin2 x−sin x−1=0

9) sin θ−cosθ=1 10) sin 2 xcos x+cos2 x sin x= 1

√2

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Key Functions: Logarithm

The graph of is shown below. Use this graph to draw the graph of

.

*Label five ordered pairs on both graphs.

*Use interval notation to list the following for the logarithmic function:

Domain:_________________________

Range:___________________________

*What is the asymptote of the graph of

? ___________________

*What is the base in ? __________

What is it called? _______________________*What is the base in ? _____________

What is it called? _________________________

1. Sketch the graph of each function. Draw and label the asymptote.

a)

b)

c)

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d)

2. Rewrite each logarithm in exponent form.

a)

b)

3. Rewrite each exponent in logarithm form.

a)

b)

4. Solve each equation for all values of ‘x’

where .

a)

b)

5. Evaluate where :

a)

b)

6. Use log properties to rewrite. Simplify if possible.

For any positive numbers M, N, a ( ), and any real number p:

a)

b)

c)

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d)

7. Solve.

a)

b)

Sec 7.4: Vectors, Operations, and Dot Product

Vector Notation and FactsOP OP

O is the initial point whereas P is the terminal point

OP ≠ PO

The magnitude of the vector OP is |OP|. |OP|=|PO|

Two vectors are equal iff (if and only if) the have the same direction and same magnitude.Same Vectors

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DefnA vector is a physical quantity that has both magnitude and direction. Typical vector quantities include velocity, acceleration, and force. We use a directed line segment to represent a vector quantity. The length of the vector represents the magnitude and the direction of the vector, indicated by the arrowhead, represents the direction of the quantity.

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The sum of two vectors is also a vector. We use the parallelogram rule to find the sum of two (geometric) vectors. Add “tail to tip”.

Add vector A with vector B. The sum A+B is called the resultant vector.

Is vector addition commutative? YES NO

For every vector v, there is a vector − v that has the same magnitude as v but opposite direction.

Draw v ,−v , 2 v ,u−v , and v−v .Ex 1 Use the vectors in the figure to graph the vector.(#14) 2 u−3 v+ w

Ex:

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DefnA vector with its initial point at the origin is called a position vector. A position vector u with its endpoint at the point (a , b ) is written ⟨ a . b ⟩. So, u=⟨a ,b ⟩.

a=¿horizontal component and b=¿ vertical component

Geometrically, a vector is a directed line segment whereas algebraically, it is like an ordered pair (whose initial point is at the origin).

The positive angle formed between the positive x-axis and a position vector is called the direction angle.

Magnitude and Direction Angle of a Vector ⟨ a ,b ⟩The magnitude (length) of vector u=⟨a ,b ⟩ is given by the following.

|u|=√a2+b2

The direction angle θ satisfies

tanθ= ba

,a≠ 0

w

v

u

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Ex 2 Given vectors u and v, find (a) 2 u (b) 2 u+3v (c) v−3 u(#26) (#28)

u=−i+2 j , v=i− j u=⟨−2 ,−1 ⟩ , v= ⟨−3 , 2 ⟩

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Magnitude and Direction Angle of a Vector ⟨ a ,b ⟩The magnitude (length) of vector u=⟨a ,b ⟩ is given by the following.

|u|=√a2+b2

The direction angle θ satisfies

tanθ= ba

,a≠ 0

Horizontal and Vertical ComponentsThe horizontal and vertical components, respectively, of a vector u having magnitude |u| and direction angle θ are the following.

a=|u|cosθ∧b=|u|sin θ

That is, u=⟨a ,b ⟩= ⟨|u|cosθ ,|u|sin θ ⟩Parallelogram Properties

1. A parallelogram is a quadrilateral whose opposite sides are parallel.2. The opposite sides and opposite angles of a parallelogram are equal, and adjacent angles of a

parallelogram are supplementary.3. The diagonals of a parallelogram bisect each other, but they do not necessarily bisect the angles of

the parallelogram.

Vector OperationsLet a ,b , c , d , and k represent real numbers.

⟨ a , b ⟩+ ⟨c , d ⟩= ⟨a+c , b+d ⟩k ∙ ⟨a ,b ⟩= ⟨ka , kb ⟩ (k is a called a scalar)If u=⟨a ,b ⟩, then –u=⟨−a ,−b ⟩

⟨ a , b ⟩−⟨ c , d ⟩=⟨ a , b ⟩+(−⟨c , d ⟩ )=⟨ a−c , b−d ⟩

DefnA unit vector is a vector that has magnitude 1. Two useful vectors are i=i= ⟨1,0 ⟩ and j= j= ⟨0,1 ⟩ and are graphed below.

If v=⟨ a ,b ⟩, then v=a i+b j.

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The dot product of two vectors is a scalar (a real number), NOT a vector. It is also known as the inner product. Dot products are used to determine the angle between two vectors, to derive geometric theorems, and to solve physical problems. (Once we have the formula for the geometric interpretation of the dot product, we will see that we can think of the dot product u ∙ v as how much the projection of u is going in the same direction of v.)

Derive the Formula for the Geometric Interpretation of the Dot Product (for 2-dim) (using Law Cosines)

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Dot ProductThe dot product of the two vectors u=⟨a ,b ⟩ and v=⟨ c , d ⟩ is denoted ∙ v , read “u dot v” and is given by the following.

u ∙ v=ac+bdNote: The dot product is a scalar, not a vector!

Properties of the Dot ProductFor all vectors u , v , and w and scalars k , the following hold.

(a) u ∙ v=v ∙ u (b) u ∙ (v+w )=u ∙ v+u ∙ w

(c) (u+v ) ∙w=u ∙ w+v ∙w (d) (k u ) ∙ v=k (u ∙ v )=u∙ (k v )

(e) 0 ∙ u=0 (f) u ∙u=|u|2

Geometric Interpretation of the Dot ProductIf θ is the angle between the two nonzero vectors u and v, where 0 ° ≤θ ≤ 180° , then the following holds.

cosθ= u ∙ v|u||v|

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Ex 3 (#30) For the pair of vectors with angle θ between them, sketch the resultant.|u|=8 ,|v|=12 , θ=20 °

Ex 4 (#34) Find the magnitude and direction angle for the vector ⟨−7 , 24 ⟩.

Ex 5 (#42) Find the magnitude of the horizontal and vertical components of v, if θ is the direction angle. θ=146.3° ,|v|=238

Ex 6 Write each vector in the form ⟨ a ,b ⟩.(#44) (#48)

Ex 7 (#52) Two forces of 37.8 lb and 53.7 lb act at a point in the plane. The angle between the two forces is 68.5 °. Find the magnitude of the resultant force.

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Geometric Interpretation of the Dot ProductIf θ is the angle between the two nonzero vectors u and v, where 0 ° ≤θ ≤ 180° , then the following holds.

cosθ= u ∙ v|u||v|

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Ex 8 (#56) Find the magnitude of the resultant force using the parallelogram rule.

Ex 9 (#66) Given u=⟨−2,5 ⟩ and v=⟨ 4,3 ⟩, find 2 u+v−6v .

Ex 10 Find the dot product for each pair of vectors.(#74) (#76)

⟨7 ,−2 ⟩ , ⟨4,14 ⟩ 2 i+4 j ,− j

Ex 11 (#78) Find the angle between ⟨1,7 ⟩ and ⟨1,1 ⟩.

Ex 12 Determine whether each pair of vectors is orthogonal.(#88) (#90) (#92)

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⟨1,1 ⟩ , ⟨1 ,−1 ⟩ ⟨ 3,4 ⟩ , ⟨ 6,8 ⟩−4 i+3 j ,8 i−6 j

Section 7-5 Application of Vectors (lecture examples)Read each problem carefully and start with a sketch of the situation. Then use what you know about vectors to solve each problem.

1) An arrow is shot into the air so that its horizontal velocity is 25 feet per second and its vertical velocity is 15 feet per second. Find the velocity of the arrow.

2) A boat is crossing a river that runs due north. The heading of the boat is due east, and it is moving through the water at 12.0 mph. If the current of the river is a constant 3.25 mph, find the true course of the boat.

3) An airplane has a velocity of 400mph southwest. A 50mph wind is blowing from the west. Find the resultant speed AND direction of the plane.

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4) A plane is flying at 170 miles per hour with heading 52.5 due north. The wind currents are a constant 35.0 miles per hour at 142.5 due north. Find the ground speed and true course of the plane.

5) Two rescue vessels are pulling a broken-down motorboat toward a boathouse with forces of 840 lb and 960 lb. The angle between these forces is 24.5. Find the magnitude and the direction of the equilibrant force with the 840 lb force.

6) Find the force required to keep a 2000-lb car parked on a hill that makes and angle of 30 with the horizontal.

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Sec 7.5: Applications of VectorsEx 1 (#8) Find the force required to keep a 3000-lb car parked on a hill that makes an angle of 15 ° with the horizontal.

Ex 2 (#21) An airline route from San Francisco to Honolulu is on a bearing of 233.0 °. A jet flying at 450 mph on that bearing encounters a wind blowing at 39.0 mph from direction of a 114.0°. Find the resulting bearing and ground speed of the plane.

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Ex 3 (#25) A pilot is flying at 190.0 mph. He wants his flight path to be on a bearing of 64 ° 30 '. A wind is blowing from the south at 35.0 mph. Find the bearing he should fly, and find the plane’s ground speed.

Ex 4 (#29) An airplane is headed on a bearing of 174 ° at an airspeed of 240 km per hr. A 30-km-per-hr wind is blowing from a direction of 245 °. Find the ground speed and resulting bearing of the plane.

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Sec 8.1: Complex Numbers

Cycle of i

Ex 1 Identify each number as rational (real), irrational (real), pure imaginary (complex), or nonreal complex. (More than one description may apply.)

−43

00 e+13 i−7 i5+i

−6−2i π √24 √−25 6−√−36

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Property of Complex ConjugatesFor real numbers a and b,

(a+bi ) (a−bi )=a2+b2

Arithmetic with Complex NumbersFor complex numbers a+bi and c+di,

(a+bi )+ (c+di )=(a+c )+ (b+d ) iand

(a+bi )−(c+di )=( a−c )+(b−d ) iWe add/subtract the real parts and add/subtract the imaginary parts.

(a+bi ) (c+di )=(ac−bd )+(ad+bc ) i

Defns The imaginary unit i is equal to √−1. i=√−1 and therefore i2=−1

A complex number has the form a+bi where a and b are real numbers. a is the real part whereas b is the imaginary part.

Math 335

Ex 2 Solve.(#30) (#36)

2 x2+3 x=−2 x2+2=2 x

Ex 3 Simplify. Write answers in a+bi form (standard form) when possible.(#22) (#24) (#38)

√−500−√−80 √−17 ∙√−17

(#40) (#48) (#52)

√−5 ∙√−15 √−12 ∙√−6√8

20+√−82

(#56) (#62) (#74)

(4−i )+(8+5i ) 3√7−( 4√7−i )−4 i+(−2√7+5 i ) ( √2−4 i ) (√2+4 i )

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(#78) (#80) (#84)

−5 i ( 4−3i )2 (3−i ) (3+i ) (2−6 i ) 4−3 i4+3 i

(#92) (#102) (#104)59 i

i−14 1i−12

Ex 4 (#107) Show that α is a square root of i.

α=√22

+ √22

i

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Sec 8.2: Trigonometric (Polar) Form of Complex NumbersWe cannot order the complex numbers (as we can with R) but ∃ a 1-1 correspondence between the set of all complex numbers, C, and the set of all 2-dimensional vectors (2 x1 matrices), R2.

Plot several complex numbers.

Plot a few complex numbers and add them.

What is the relationship between the sum of complex numbers and the sum of vectors?

What is the rectangular form of a complex number?

Establish relationships among x , y , r , and θ using the figure.

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x+ yi

Relationships Among x , y , r , and θ

1)

2) 4)

3)

x+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yix+ yi

Math 335

What is the polar form of a complex number?

Ex 1 Find the sum of the complex numbers. Graph each and their resultant.(#24)

2+3 i∧−4+2 i−15+ 2

7i∧3

7− 3

4i

Ex 2 Write each complex number in rectangular form.(#34) (#38)

6 cis 135 ° √2 (cos (−60 ° )+i sin (−60° ) )

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Polar Form of a Complex NumberThe polar (trigonometric) form of the complex number z=x+ yi is

z=r (cosθ+ isin θ )

where r=|z| and θ is the angle direction angle of the vector ⟨ x , y ⟩. r is called the modulus of z and θ is called the argument of z. We often choose θ∈ [0 , 360 ° ). Abbreviation: r (cosθ+i sin θ )=r cisθ

Math 335

Ex 3 Write each complex number in polar form with θ∈ [0 ,360 ° ). (#40) (#44) (#48)

1+i √3−2+2i−2i

Ex 4 (#59-62) Describe the graphs of all complex numbers z satisfying the specified conditions.

The absolute value of z is 1. The real and imaginary parts of z are equal.

The real part of z is 1. The imaginary part of z is 1.

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Sec 8.3: The Product and Quotient Theorems

Find z1 z2 where z1=r1 cis θ1 and z2=r2 cis θ2.

Find where z1=r1 cis θ1

and z2=r2 cis θ2

.

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Product TheoremIf r1 (cosθ1+i sinθ1 ) and r2 (cosθ2+i sin θ2 ) are any two complex numbers, then the following holds.

r1 (cosθ1+i sinθ1 )∙ r2 (cosθ2+i sin θ2 )=r 1r2 [cos (θ1+θ2)+ isin (θ1+θ2 ) ]

In compact form: (r1 cis θ1 ) (r2 cisθ2 )=r1 r2 cis ( θ1+θ2 )

Multiply the lengths. Add the angles.

Quotient TheoremIf r2 (cosθ2+i sin θ2 )≠ 0, then the following holds.

r1 ( cosθ1+isin θ1 )r2 ( cosθ2+isin θ2)

=r1

r2[ cos (θ1−θ2 )+isin (θ1−θ2 ) ]

In compact form: r1 cisθ1

r2 cisθ2=

r1

r2cis (θ1−θ2 )

Math 335

Ex 1 Find each and write in rectangular form.(#8) (#12)

[8 (cos210 °+isin 210 ° ) ] [2 (cos330 °+i sin330 ° ) ] (3 cis 300° ) (7 cis 270 ° )

(#16)

12 ( cos23°+i sin 23 ° )6 (cos 293 °+ isin 293 ° )

(#20) (#24)

2 i−1−i √3

−3√2+3 i√6√6+i √2

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Product TheoremIf r1 (cosθ1+i sinθ1 ) and r2 (cosθ2+i sin θ2 ) are any two complex numbers, then the following holds.

r1 (cosθ1+i sinθ1 )∙ r2 (cosθ2+i sin θ2 )=r 1r2 [cos (θ1+θ2)+ isin (θ1+θ2 ) ]

In compact form: (r1 cis θ1 ) (r2 cisθ2 )=r1 r2 cis ( θ1+θ2 )

Multiply the lengths. Add the angles.

Quotient TheoremIf r2 (cosθ2+i sin θ2 )≠ 0, then the following holds.

r1 ( cosθ1+isin θ1 )r2 ( cosθ2+isin θ2)

=r1

r2[ cos (θ1−θ2 )+isin (θ1−θ2 ) ]

In compact form: r1 cisθ1

r2 cisθ2=

r1

r2cis (θ1−θ2 )

Math 335

Sec 8.4: De Moivre’s Theorem; Powers and Roots of Complex Numbers

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nth Root TheoremIf n is natural number, r is a positive real number, and θ is in degrees, then the nonzero complex number r (cosθ+i sin θ ) has exactly n distinct nth roots given by the following.

n√r (cosα +isin α )∨n√r cis αwhere

α=θ+360 ° kn

∨α=θn+ 360 ° k

n, k=0 ,1, 2 , …, n−1

If θ is in radians, then

α=θ+2 πkn

∨α=θn+ 2 πk

n, k=0 , 1 ,2 , …,n−1

DefnFor a natural number n, the complex number a+bi is an nth root of the complex number x+ yi if

(a+bi )n=x+ yi

De Moivre’s TheoremIf z=r (cosθ+ isin θ ) is a complex number and if n is any natural number then the following holds.

[r (cosθ+i sin θ ) ]n=r n (cos nθ+isin nθ )

In compact form:(r cis θ )n=rn ( cis nθ )

Math 335

Proof of nth Root Theorem:

Ex 1 Find each power. Write in rectangular form.(#4) PP (#6)

[2 (cos120 °+ isin 120 ° ) ]3 [ 3cis 40 ° ]3Ans: 8

(#10)

(√22

−√22

i)8

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Math 335

Ex 2 (a) Find all cube roots of each complex number. Leave answers in polar form.(b) Graph each cube root as a vector in the complex plane.(#20)

27

Ex 3 (#30) Find and graph all fourth roots of i.

Ex 4 Solve for all (complex) solutions. Leave answer in rectangular form.x3+1=0

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Try problems:

Sec 8.5: Polar Equations and GraphsPage 166 of 186

Math 335

Why polar coordinates? Different, sometimes more efficient, way to represent a point or graph. (e.g. shortest distance) Makes some applications/equations easier to work with (e.g. r=3 vs ____________________

and using DeMoivre's Theorem vs Binomial Theorem) Some non-functions in rectangular coordinates are functions in polar coordinates so we can

apply rules/theorems of functions that we wouldn't be able to otherwise

Rectangular Coordinates VS Polar Coordinates

Pole Polar Axis Coordinates of the pole ___________

Examples: (4 ,270° ) (1 , π6 ) (−3 , 120 ° ) (−2 ,− π

2 )

Are points uniquely defined?

Ex 1 Find the rectangular coordinates of each point.

(−3 , π ) (−2 , 2 π3 ) (−2 ,− π

4 )

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Polar Coordinates Rectangular Coordinates(r , θ ) ↔ ( x , y )

x=r cosθ y=r sinθ r 2=x2+ y2 tan θ= yx

, x ≠ 0

Math 335

Ex 2 Find the polar coordinates of each rectangular coordinate. (0 ,−2 ) (−3 , 3 ) (−2 ,−2√3 )

Ex 3 Rewrite each rectangular equation in polar equation.

x2+ y2=x x2=4 y x=4 y=−3

Ex 4 Rewrite each polar equation in rectangular equation.

r=sin θ r=2 r= 33−cosθ

Ex 5 Transform each polar equation into rectangular equation. Then identify and graph the equation.Page 168 of 186

Symmetry WRT…

Polar axis Replace θ by −θ. Line θ=π2 Replace θ by π−θ Pole Replace r by −r

Math 335

r=2 θ=−π4 r=2sin θ

Lines Should be able to convert to polar form. Need to know the graphs of polar form for all but y=mx+b type.

Polar Form Graphy=5

y=−2

x=4

x=−3

y=mx ( y=2 x)

y=mx+b -------------------------- --------------------------

ax+by=c

Circles Rectangular Form Graph

r=3

r=2sin θ

Rose Curves

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r=4 cos 3θr=4sin 3θ3 petals

r=4cos 2θr=4sin 2θ4 petals

r=−4 cos 3θr=−4 sin3 θ3 petals

r=−4 cos2θr=−4 sin2θ4 petals

(www.wolframalpha.com)

NOTE: Cosine rose curves start on _________________ and Sine rose curves start in __________ and has a petal on the __________________-__________.

Cardioid Passes through pole

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r=5cos3θ r=5sin3θ r=5cos2θr=5sin2 θr=5cos5θ r=5sin5 θ r=5cos4θ

Math 335

r=2+2cos θr=2−2cosθ

r=2+2sin θr=2−2 sin θ

Limaçon w/out Inner Loop Does NOT pass through pole

r=3+2cosθr=3−2 cosθ

r=3+2 sinθr=3−2sin θ

Limaçon w/Inner Loop Passes through pole

r=1+2cosθr=1−2cosθ

r=1+2sin θr=1−2 sin θ

Lemniscate Figure 8

r2=4sin (2θ )

r2=4 cos (2θ )

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The Cycle of a Rose Curve

For each Rose Curve, fill in enough of the table to complete the graph one time.

1)

0

r

2)

0

r

3)Page 173 of 186

Math 335

0

r

4)

0

r

Name :____________________________

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Polar Equations and Graphs

1. 2.

3. 4.

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5. 6.

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7. 8.

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Math 335 Polar Equations and Graphs Homework Handout

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Use pencil and circle final answers.

Convert each rectangular equation into polar equation. (solve for ‘r =’ whenever appropriate)

1) 2)

3) 4)

5) 6)

7) 8)

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Math 335

Convert each polar equation into rectangular equation.

9) 10)

11) 12)

13) 14)

Problems 15 – 24. Graph each polar equation.

15) 16)

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Math 335

17) r = 2 18)

19) 20)

21) 22)

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23) 24)

Problems 25 – 26. Graph each of the polar equations.25)

0

r

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26)

0

r

27. Circle the graph that matches the given polar equation.(a) (b)

(c) (d)

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28.

a. b.

c. d.

29.

a. b.

c. d.

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30.

a. b.

c. d.

Graph each polar equations.31. 32.

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33. 34.

35. 36.

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marcolsen.weebly.com · Web viewWord Problems Practice 49 2.5 Further Applications of Right Triangles 51 7.1 Oblique Triangles and The Law of Sines 53 7.3 The Law of Cosines 59 7.2