Advanced Algebra w Trig (Grades 11-12) · Advanced Algebra w Trig (Grades 11-12) 12 Review In studying the graphs of sine and cosine, many important ideas surfaced: period, periodic

Advanced Algebra w Trig (Grades 11-12)

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CHARLES COUNTY PUBLIC SCHOOLS

Advanced Algebra with Trig (Grades 11-12) Mathematics

Weeks 7-8 (May 18 – May 29)

Dear parents,

If your child is participating in distance learning solely through the completion of our instructional packets, you are required to call or email the principal to inform them of your child’s participation status, since packet-assignments will not be collected until a later time. Please keep all of your child’s work in a safe place until you are notified of when, where and how to submit. Thank you for your attention to this matter.


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Estimados padres, Si su hijo/a está participando en el aprendizaje a distancia completando solamente nuestros paquetes de instrucción, deberá llamar o enviar un correo electrónico al director para informarle sobre el estado de participación de su hijo/a, ya que las asignaciones realizadas en los paquetes no se recopilarán hasta más tarde. Por favor mantenga todo el trabajo de su hijo/a en un lugar seguro hasta que se le notifique cuándo, dónde y cómo presentarlo. Gracias por su atención a este asunto.


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Student: _________________________________ School: _____________________________

Teacher: _________________________________ Block/Period: ________________________

Packet Directions for Students Week 7:

Read through the Instruction and examples on Graphs of Sine and Cosine while completing the corresponding questions on 11.4.1 Study: Graphs of Sine and Cosine worksheet

Complete 11.4.1 Study: Graphs of Sine and Cosine o Check and revise solutions using the 11.4.1 Study: Graphs of Sine and Cosine

Answer Key

Complete Quiz: Graphs of Sine and Cosine

Week 8:

Read through the Instruction and examples on the Graphs Other Functions while completing the corresponding questions on the 11.5.1 Study: Graphs of Other Functions

o Complete 11.5.1 Study: Graphs of Other Functions o Check and revise solutions using the 11.5.1 Study: Graphs of Other Functions

Answer Key

Complete Quiz – Graphs of Other Functions


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Graphs of Sine and Cosine The previous unit taught you how to determine values of the sine and cosine functions by using your knowledge of right triangles from geometry.

The graph of a function is made up of all ordered pairs (x,f (x)). Recall that the sine and cosine functions are defined for any angle x. In this lesson, we will pair angles with the corresponding values of the sine or cosine functions to generate their graphs. It is important to remember that the graph of a function y = f(x) is defined as the set of all ordered pairs {(x,f(x))}. Thus, to sketch these graphs, you just have to plot points in the x-y plane. The graphs of sine and cosine are examples of a family of curves called sinusoids. Graphs of Sine and Cosine

Use critical points to sketch the graphs of the functions sine and cosine. Describe the domain and range of the functions sine and cosine. Understand and use the periodic nature of the functions sine and cosine to sketch

complete graphs of these functions. Recognize graphically if a function is even or odd.

What Is a Sinusoid, Anyway?

A schematic of the Antikythera mechanism, an ancient device used to calculate astronomical positions


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The family of curves called sinusoids are based on the graph of the trigonometric function sine (or cosine).

Sinusoids were used in ancient civilizations as a tool for indirect measurements and were linked heavily to right triangles. Hindu mathematicians used the sine ratio to solve astronomy problems, and this knowledge also appears to have been shared by Greek mathematicians.

Once the concept of a function was introduced, especially as it was formalized by the field of calculus in the 1700s, the trigonometric functions moved past their roots in triangles and measurement.

Graphing Sine

One way to visualize the graph of a sine curve is to relate each point on the curve to a point on the unit circle. As the angle passes through all possible values, you can imagine the curve being traced out. How do you do that? It is important to remember how you can define the trigonometric functions using the unit circle. Does the diagram above look familiar? This should remind you that the x-coordinate of a point on the unit circle is equal to cos , and the y-coordinate is equal to sin . You can use this property to get a picture of the graph of sine.


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Plotting Points Now that you have used the circle to see what the graph of sin looks like, it is time to generate the graph of the sine function the old-fashioned way — by completing a table of values and plotting them on a graph.


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Sine as a Function

Here is a summary of what you have just seen, along with a few more interesting facts about this graph.

Pay particularly close attention to the new terms odd function and a periodic function with period p that

are introduced here.


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Graphing Cosine

You can use the same process to graph the function cos . Again, you will need to use the fact that the x-

coordinate of a point on the unit circle is equal to cos . As you will see, the graph of cos will share

many of the same properties as the graph of sin . This should not be surprising at all if you consider the

symmetry of the unit circle.

Plotting Points, AgainNow that you have used the circle to get an idea of what the graph of cos looks

like, it is time to generate the graph of the cosine function the old-fashioned way — by completing a

table of values and plotting them on a graph.


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Cosine as a Function

Here is a summary of what you have just seen, along with a few more interesting facts about this graph.

Note that in contrast to the sine function, which is an odd function, cosine is an even function.


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A Note on Periodic Functions

You have just seen that both sine and cosine are periodic functions with period . What other

functions are periodic?

It turns out that all six trigonometric functions are periodic, but there are a lot more than that!

The Five Essential Points

Once you are comfortable with the basic shapes of the sine and cosine functions, you can actually sketch

the graph of these functions with far fewer points.

The essential points you should plot are the zeros of the function, as well as the maximum and minimum

values. This gives a total of five points for one period.

One period of a sinusoid with the five essential points plotted

Since these graphs are periodic, all you need to do is sketch one period. The rest of the graph is just that

period repeated again and again. Remember: It is important to sketch the graph as a smooth curve — no

sharp corners on those hills


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A Note on Notation

You might have noticed that in the previous example, the independent variable was replaced by the

independent variable x.

It is important to be comfortable with both ways of notating the independent variable since both have

their benefits.

Using is beneficial since it serves as a reminder that the trigonometric functions were defined in terms

of the unit circle.

Is there any difference?

As you begin to study the graphs and function properties of the trigonometric functions, the

notation x is convenient since the general notation for a function is f(x). This enables you to graph these

functions in the xy-plane rather than the y-plane.

In the end, it all amounts to perspective, and it does not really matter which variable you use.


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Review

In studying the graphs of sine and cosine, many important ideas surfaced: period, periodic behavior, odd

and even functions, and just general graphical recognition of sine and cosine. Familiarity with these

ideas will be important as you continue your studies of trigonometry.

periodic function

A function, such as sin x, whose value is repeated at constant intervals

odd function

A function ƒ(x) is odd if, for every x, ƒ(-x) = -ƒ(x)

even function

A function with the property that ƒ(x) = ƒ(-x) for each number x.

Satellites

Think about some of the conveniences of modern life. Cell

phones keep us in touch with loved ones, and televisions bring news from around the world instantly

into our homes. Weather reports let us know when it's going to be sunny, and GPS lets us enjoy taking

trips without the fear of getting lost. All of these things rely on satellites in orbit around Earth.

Most satellites fly in a circular orbit high above the Earth. Because their paths are circular and periodic,

the motion of these satellites can be represented by a sinusoid.

But who actually uses the graphs of those sinusoids? Well, funny you should ask! Antennas here on

Earth watch those satellites carefully, and if the actual position of a satellite begins to differ from what

the graph predicts, powerful computers send instructions to the satellite to correct the error and keep it

from falling out of orbit (and losing your GPS signal).


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8.1.1 Study: Graphs of Sine and Cosine

Name:

Date:

Use the questions below to keep track of key concepts from this lesson's study activity.

Page 1:

The family of curves called __________ are based on the graph of the trigonometric function sine (or

cosine).

Pages 2 – 4:

a. Graph the sine function on the coordinate grid below.

b. The domain of the sine function is ____________________.

c. The range of the sine function is __________.

d. The sine function is __________, meaning it has symmetry about the

__________.

e. The period of the sine function is _____.


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Pages 5 – 7:

a. Graph the cosine function on the coordinate grid below.

b. The domain of the cosine function is ____________________.

c. The range of the cosine function is __________.

d. The cosine function is __________, meaning it has symmetry about the

__________.

e. The period of the cosine function is _____.

Page 8:

On the coordinate grid below, draw the graph of a periodic function other than the sine or cosine

function.


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Pages 9 – 10:

a. For the function , x is the __________ variable, and y is the __________ variable.

b. What is the domain of ?

c. What is the period of ?

d. What is the range of ?

e. Graph the function on the coordinate grid below.

Page 12: The following is the graph of what function?


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8.1.1 Study: Graphs of Sine and Cosine

ANSWER KEY

Page 1:

The family of curves called __________ are based on the graph of the trigonometric function sine (or

cosine).

sinusoids

Pages 2 – 4:

a. Graph the sine function on the coordinate grid below.

The graph should appear as follows.

b. The domain of the sine function is ____________________.

all real numbers

c. The range of the sine function is __________.

d. The sine function is __________, meaning it has symmetry about the

__________.

odd; origin

e. The period of the sine function is _____.


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Pages 5 – 7:

a. Graph the cosine function on the coordinate grid below.


b. The domain of the cosine function is ____________________.

all real numbers

c. The range of the cosine function is __________.

d. The cosine function is __________, meaning it has symmetry about the

__________.

even; y-axis

e. The period of the cosine function is _____.

Page 8:

On the coordinate grid below, draw the graph of a periodic function other than the sine or cosine

function. Answers will vary. One example is the following graph.


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Pages 9 – 10:

a. For the function , x is the __________ variable, and y is the __________ variable.

independent; dependent

b. What is the domain of ?

all real numbers

c. What is the period of ?

d. What is the range of ?

e. Graph the function on the coordinate grid below.


Page 12:

The following is the graph of what function?


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Quiz: Graphs of Sine and Cosine Question 1a of 10

A sinusoid is a function whose values repeat based on positions of a point that moves around a circle.

A. True

B. False

Question 2a of 10

The domain of the sine function is _____.

A. all real numbers

B.

C.

D. [-1,1]

Question 3a of 10

Which of the following functions is not a sinusoid?

A. y = |x|

B. y = sin x

C. y = cos x

D. None of the above are sinusoids.


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Question 4a of 10

Which graph or graphs appear to show a sinusoid?

A. I only

B. III only

C. I and II only

D. II only

Question 5a of 10

Which function's graph is shown below?

A. y = -sin x

B. y = -cos x

C. y = cos x

D. y = sin x


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Question 6a of 10

Which natural phenomenon is the best example of periodic behavior?

A. The closing value of the stock market at the end of each day

B. The number of fish in a pond as a function of time

C. The amount of pollution in Los Angeles as a function of time

D. The number of hours of daylight each day

Question 7a of 10

What is the period of the function y = 2sin x?

A.

B. All real numbers

C.

D. [-1,1]

Question 8a of 10

What is the range of the function y = 2sin x?

A. [-2,2]

B.

C.

D. All real numbers


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Question 9a of 10

The cosine function is an odd function.

# Choice

A. True

B. False

Question 10a of 10

What is the minimum number of points required to mark all maximum, minimum, and zeros in a period

of a sinusoid?

Answer:


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Graphs of Other Functions

As we have begun to explore them, we have seen that trigonometric functions can be graphed like any

other mathematical function. We can go further than just the sin and cos functions though.

Recall the definition of the tangent function based on the unit circle. It can be given as a ratio of sine and

cosine functions or as a ratio of the legs of a right triangle with angle :

If the tangent function is a ratio of the sine and cosine functions, how do you think the graph will

behave? Will it be a periodic function?


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Graphing Cotangent

The cotangent function is closely related to the tangent function. Recall the definition of the

cotangent function based on the unit circle. It can be given as a ratio of cosine and sine functions or as a

ratio of the legs of a right triangle with angle :

If the cotangent function is a ratio of the cosine and sine functions, how do you think the graph will

behave? Will it be periodic as the tangent function was?


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Cotangent Summary

Here is a summary of what you have just seen along with a few more interesting facts about this graph.

Graphing Secant

Recall the definition of the secant function based on the unit circle:

The secant function is the inverse of cosine, or 1 divided by the length of the horizontal leg of the right

triangle formed by angle .


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Secant Summary



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Graphing Cosecant

Recall the definition of the cosecant function based on the unit circle:

The cosecant function is the inverse of sine, or 1 divided by the length of the vertical leg of the right

triangle formed by angle . As the inverse of the sine function, how do you think the graph of cosecant

will behave? Will it be a periodic function?


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Cosecant Summary



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Graphing with a Little Help

If you have a hard time memorizing all the shapes of the graphs for trigonometric functions, it might

help you to think about their relationships to the sine and cosine functions.

Knowing only the shapes of the sine and cosine graphs, along with how they relate to the other

trigonometric functions, you can draw the graphs of all the other trig functions.


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Summarizing the Trigonometric Functions

It's time to organize the information you have about these four graphs into a handy reference guide for

your notebook. Grab a sheet of paper and follow these steps.

Fold the paper in half lengthwise (a "hot dog" fold).

Unfold the paper and fold it in half the other direction (a "hamburger" fold).

Unfold the paper and turn it so that the long edge is parallel to you.

Fold the ends in until they meet at the middle crease.

Cut flaps so that you have four flaps opening from the center outward, revealing four rectangles

inside.

Write the name of the four functions from this lesson on the outside of each flap.

On the underside of each flap, write the domain and range, as well as any other information you

think is helpful.

On the inner rectangle that is revealed behind the flap, sketch the graph.


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8.2.1 Study: Graphing More Trigonometric Functions

Name:

Date:

Use the questions below to keep track of key concepts from this lesson's study activity.

Pages 1 – 2:

a. Graph the tangent function on the coordinate grid below.

b. The domain of the tangent function is ____________________.

c. The range of the tangent function is ____________________.

d. The tangent function is __________, meaning it has symmetry about the __________.

e. The period of the tangent function is _____.

Pages 3 – 4:

a. Graph the cotangent function on the coordinate grid.

b. The domain of the cotangent function is _________ .

c. The range of the cotangent function is ___________.

d. The cotangent function is __________,

meaning it has symmetry about the __________.

e. The period of the cotangent function is _____.


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Pages 5 – 6:

a. Graph the secant function on the coordinate grid.

b. The domain of the secant function is _______________.

c. The range of the secant function is _________________.

d. The secant function is __________,

meaning it has symmetry about th__________.

e. The period of the secant function is _____.

Pages 7 – 8:

a. Graph the cosecant function on the coordinate grid.

b. The domain of the cosecant function is ________________.

c. The range of the cosecant function is __________________.

d. The cosecant function is __________,

meaning it has symmetry about the__________.

e. The period of the cosecant function is _____.

Page 9:

a. Whenever , is __________.

b. Whenever , is __________.

c. Whenever , is __________.

d. Whenever is positive, is __________.

e. Whenever is negative, is __________.

f. Whenever , is __________.

g. Whenever , is __________.

h. Whenever and have the same sign, is __________.

i. Whenever and have opposite signs, is __________.


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8.2.1 Study: Graphing More Trigonometric Functions

ANSWER KEY

Pages 1 – 2:

a. Graph the tangent function on the coordinate grid.


b. The domain of the tangent function is ____________________.

c. The range of the tangent function is ____________________.

all real numbers

d. The tangent function is __________, meaning it has symmetry about the __________.

odd; origin

e. The period of the tangent function is _____.


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Pages 3 – 4:

a. Graph the cotangent function on the coordinate grid.


b. The domain of the cotangent function is ____________________.

c. The range of the cotangent function is ____________________.

all real numbers

d. The cotangent function is __________, meaning it has symmetry about the __________.

odd; origin

e. The period of the cotangent function is _____.


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Pages 5 – 6:

a. Graph the secant function on the coordinate grid below.


b. The domain of the secant function is ____________________.

c. The range of the secant function is ____________________.

d. The secant function is __________, meaning it has symmetry about the __________.

even; y-axis

e. The period of the secant function is _____.


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Pages 7 – 8:

a. Graph the cosecant function on the coordinate grid below.


b. The domain of the cosecant function is ____________________.

c. The range of the cosecant function is ____________________.

d. The cosecant function is __________, meaning it has symmetry about the __________.

odd; origin

e. The period of the cosecant function is _____.


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Page 9:

a. Whenever , is __________.

undefined

b. Whenever , is __________.

1

c. Whenever , is __________.

-1

d. Whenever is positive, is __________.

positive

e. Whenever is negative, is __________.

negative

f. Whenever , is __________.

undefined

g. Whenever , is __________.

0

h. Whenever and have the same sign, is __________.

positive

i. Whenever and have opposite signs, is __________.

negative


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Quiz: Graphs of Other Functions Question 1

1a ) Choose the function whose graph is given below.

A. y = cot x

B. y = csc x

C. y = tan x

D. y = sec x

Question 2


A. y = cot x

B. y = sec x

C. y = tan x

D. y = csc x


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Question 3


A. y = csc x

B. y = tan x

C. y = cot x

D. y = sec x

Question 4


A. y = cot x

B. y = sec x

C. y = tan x

D. y = csc x


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Question 5

5a ) Which of the following describes the domain of y = tan x, where n is any integer?

A.

B.

C.

D.

Question 6

6a ) The domain of y = csc x is given by .

A. True

B. False

Question 7

7a ) The vertical _____ of the function secant are determined by the points that are not in the

domain.

Answer:


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Question 8

8a ) Which of the functions below have period ? Check all that apply.

A. y = sec x

B. y = cot x

C. y = tan x

D. y = csc x

Question 9

9a ) Which of the functions have a range of all real numbers? Check all that apply.

A. y = csc x

B. y = tan x

C. y = cot x

D. y = sec x

Question 10

10a ) Which function is even? Check all that apply.

A. y = sec x

B. y = cos x

C. y = sin x

D. y = csc x

Documents

Advanced Algebra w Trig (Grades 11-12) · Advanced Algebra w Trig (Grades 11-12) 12 Review In studying the graphs of sine and cosine, many important ideas surfaced: period, periodic