Algebra 2 (Grades 10-12) - Charles County Public Schools › images › distancelearning › week... · Algebra 2 (Grades 10-12) 8 11.3.1 Study: Trig Ratios and the Unit Circle Name:

Algebra 2 (Grades 10-12)

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CHARLES COUNTY PUBLIC SCHOOLS

Algebra 2 (Grades 10-12) 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 5:

Read through the Instruction and examples on the 11.3.1 Trig Ratios and the Unit Circle while completing the corresponding questions on the 11.3.1 Study: Trig Ratios and the Unit Circle and 11.3.2 Study: Pythagorean Theorem

Complete Study: 11.3.1 Trig Ratios and the Unit Circle o Check and revise solutions using the Study: 11.3.1 Trig Ratios and the Unit Circle

Answer Key

Complete 11.3.2 Study: Pythagorean Theorem

o Check and revise solutions using the 11.3.2 Study: Pythagorean Theorem Answer

Key

Complete Quiz: Trigonometric Functions and the Unit Circle

Week 6:

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


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Trigonometric Ratios and the Unit Circle

The trigonometric functions introduced in the last lesson are exactly what we need for modeling this

kind of change. In this lesson, you will see that the six trigonometric functions can be defined using a

unit circle — that is a circle with a radius of 1 — and you will learn to use the unit circle to find values of

the trigonometric functions for angles greater than 90 degrees (or radians).

Terminal Conditions

In this lesson, you will look at the trigonometric functions again, this time from a slightly different

perspective — using a circle. While the triangle allowed us to define the trigonometric functions for

angles between 0 and 90 degrees (or between 0 and radians), the definitions developed in this

lesson will allow us to find values of trigonometric functions for any real number.

We will see that trigonometric functions are especially useful for representing the kind of repetitive

motion seen here as this bicyclist pedals. Notice that her foot goes around and around, repeating the

same motion over and over. The trigonometric functions are sometimes called "circular functions"

because of this repeated circular behavior.

Review the Functions

In the last lesson, you were introduced to the six trigonometric functions and their relationships to the

angles and side lengths of a right triangle.


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Trigonometric Functions from the Unit Circle

As useful as trigonometric functions are in relating the sides of a right triangle with its angles, this is not

the only time that these functions prove useful. We will expand our use of trigonometric functions from

angles less than 90 degrees and radians to all possible real angle values.

To define trigonometric functions more generally, begin by looking at the unit circle.

New Definitions for Trigonometric Functions

The table below reviews the new set of definitions for the six trigonometric functions. is the angle (in

radians) determined by the terminal point on the unit circle and can be any real number. The

coordinates of the terminal point are x and y.


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The Circle, So Far

The table below summarizes the information you've found so far using the trigonometric definitions

derived from the unit circle. See if you can find any patterns in the values for each function as

angle increases around the entire circle.

(degrees)

(radians)

0 0 0 1 0 undef. 1 undef.

90

1 0 undef. 1 undef. 0

180

0 -1 0 undef. -1 undef.

270

-1 0 undef. -1 undef. 0

You are going to continue to build the unit circle by concentrating on the first quadrant — that is the

part of the circle where x- and y-values are positive. To do this, you can use what you know about the

ratios of the sides of some special right triangles.

Putting it all together, you have the coordinate locations of several more terminal points and their

corresponding angles. This will allow you to solve some trigonometric equations using their definitions.

Take a look at some examples.

Reference Angles

You have begun exploring some new definitions for common trigonometric functions. By now, you've

solved for the coordinates of a few special points on the unit circle that define 30-60-90 and 45-45-

90 triangles in the first quadrant. However, you haven't yet seen how trigonometric functions are

handled when the terminal point is located in quadrants other than the first.

Now you will learn how to use reference angles and reference points to solve for the coordinates of

terminal points on the unit circle located in the second, third, and fourth quadrants.


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Reference Angle Examples

The unit circle with reference angles

The Unit Circle from Every Angle


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11.3.1 Study: Trig Ratios and the Unit Circle

Name:

Date:

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

Page 1:

Trigonometric functions are sometimes called __________ functions.

Page 2:

Define the six trigonometric ratios for using the triangle below.

a. sin = _______________

b. cos = _______________

c. tan = _______________

d. csc = _______________

e. sec = _______________

f. cot = _______________


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Pages 3 – 4:

Define each of the six trigonometric functions when the terminal point P has the coordinates (x,y) in the

unit circle below. Assume x and y are not equal to 0.

a. sin = __________

b. cos = __________

c. tan = __________

d. csc = __________

e. sec = __________

f. cot = __________

Pages 5 – 6:

Fill in the missing information in the table below using the trigonometric definitions derived from the

unit circle.

(degrees) (radians) sin cos tan csc sec cot

0°

180°


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Page 7:


unit circle.


45°

Page 8:

Give the reference angle for each of the following angles.

a.

b.

c.

d.

e.

f.

g.

h.

i.

Pages 9 – 10:


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unit circle.


120°

150°

225°

300°

330°


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11.3.1 Study: Trig Ratios and the Unit Circle

ANSWER KEY

Page 1:

Trigonometric functions are sometimes called __________ functions.

circular

Page 2:

Define the six trigonometric ratios for using the triangle below.

a. sin = _______________

b. cos = _______________

c. tan = _______________

d. csc = _______________

e. sec = _______________

f. cot = _______________


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Pages 3 – 4:

Define each of the six trigonometric functions when the terminal point P has the coordinates (x,y) in the

unit circle below. Assume x and y are not equal to 0.

a. sin = __________

y

b. cos = __________

x

c. tan = __________

d. csc = __________

e. sec = __________

f. cot = __________


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Pages 5 – 6:


unit circle.

The table should appear as follows.


0° 0 0 1 0 undefined 1 undefined

90°

1 0 undefined 1 undefined 0

180°

0 -1 0 undefined -1 undefined

270°

-1 0 undefined -1 undefined 0

Page 7:


unit circle.



30°

2

45°

1

1

60°

2


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Page 8:

Give the reference angle for each of the following angles.

a.

b.

c.

d.

e.

f.

g.

h.

i.


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Pages 9 – 10:


unit circle.



120°

-2

135°

-1

-1

150°

2

210°

-2

225°

1

1

240°

-2

300°

2

315°

-1

-1

330°

-2


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11.3.2 Pythagoream Theorem

You can relate the Pythagorean theorem to the unit circle to see a fundamental relationship between

sine and cosine.

Proving It

You could have derived this relationship without using the Pythagorean theorem. How?

The equation of the unit circle is x2 + y2 = 1.

Remember that when we defined cos θ to be x and sin θ to be y, x and y referred to the coordinates of a

point on the unit circle.

What is the equation of the unit circle?

If you substitute cos and sin for x and y in this equation, you obtain the Pythagorean identity:

cos2 θ + sin2 θ = 1

It doesn't matter whether you remember this relationship by thinking about the Pythagorean theorem

or by thinking about the equation of the unit circle. But it does matter that you remember it!


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Confirm

You have now learned about unit circles, their trigonometric definitions, and ratios. Answer the question

below to confirm your understanding.

A unit circle is a circle with radius , with the relation between and (x, y) defined

by and .

Definition: Pythagorean identity

Here is one form:

Here are two more:

Example:

(0, -1)

Example: What's the value of the angle with reference point ?

11𝜋

6 (The angle is in the fourth quadrant with a reference angle of )

Example: What is the value of ?

1

2

Example: What is the value of ?

−√3

2 (The sine of the reference angle is . Because the angle of the problem is in the third

quadrant, this value will take a negative sign.)


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Here is a summary of what you have seen in this lesson.


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11.3.2 Study: Pythagorean Theorem

Name:

Date:

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

Pages 1 – 4:

a. What is the equation of the unit circle shown below?

b. What does the Pythagorean theorem say about the relationship between x and y?

c. List the three trigonometric identities that can be derived from the unit circle.

1. ______________________________

2. ______________________________

3. ______________________________


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11.3.2 Study: Pythagorean Theorem

ANSWER KEY

Pages 1 – 4:

a. What is the equation of the unit circle shown below?

b. What does the Pythagorean theorem say about the relationship between x and y?

c. List the three trigonometric identities that can be derived from the unit circle.

1. ______________________________

2. ______________________________

3. ______________________________

; ;


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Quiz: Trigonometric Functions and the Unit Circle Question 1a of 10

sin( ) = _____

A.

B.

C.

D.

Question 2a of 10

Check all that apply. is the reference angle for:

A.

B.

C.

D.


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Question 3a of 10

Which of the following could be points on the unit circle?

A.

B.

C.

D.

Question 4a of 10

If is the point on the unit circle determined by real number , then tan = _____.

A.

B.

C.

D.


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Question 5a of 10

If sin > 0 and cos > 0, then the terminal point determined by is in:

A. quadrant 2.

B. quadrant 3.

C. quadrant 1.

D. quadrant 4.

Question 6a of 10

If tan = and the terminal point determined by is in quadrant 3, then:

A.

sin =

B.

csc =

C.

cos =

D.

cot =


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7a. The statement "tan = , csc = , and the terminal point determined by is in

quadrant 3":

A. cannot be true because tan is greater than zero in quadrant 3.

B.

cannot be true because if tan = , then csc = .

C. cannot be true because tan must be less than 1.

D. cannot be true because .

Question 8a of 10

Check all that apply. tan is undefined for = _____.

A.

B.

C.

D. 0


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Question 9a of 10

sin( ) = _____

A.

B.

C.

D.

Question 10a of 10

cot( ) = _____

A. 0

B. -1

C. 1

D. Undefined


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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:

Documents

Algebra 2 (Grades 10-12) - Charles County Public Schools › images › distancelearning › week... · Algebra 2 (Grades 10-12) 8 11.3.1 Study: Trig Ratios and the Unit Circle Name: