apex pre-calculus learning packet · pre-calculus: weeks 3-4, april 20 – may 1 pages 3 – 4:...

Pre-Calculus: Weeks 3-4, April 20 – May 1

CHARLES COUNTY PUBLIC SCHOOLS

APEX Pre-Calculus

Learning Packet


Student: _________________________________ School: _____________________________

Teacher: _________________________________ Block/Period: ________________________

Packet Directions for Students Week 3

Read through the Instruction and examples on Trig Ratios and the Unit Circle while completing the corresponding questions on 7.3.1 Study: Trig Ratios and the Unit Circle worksheet

Complete 7.3.1 Study: Trig Ratios and the Unit Circle o Check and revise solutions using the 7.3.1 Study: Trig Ratios and the Unit Circle

Answer Key

Complete Quiz – Trig Ratios and the Unit Circle

Week 4

Read through the Instruction and examples on Graphs of Sine and Cosine while completing the corresponding questions on 8.1.1 Study: What is a Sinusoid Anyway? worksheet

Complete 8.1.1 Study: What is a Sinusoid Anyway? o Check and revise solutions using the 8.1.1 Study: What is a Sinusoid Anyway?

Answer Key

Complete Quiz – What is a Sinusoid Anyway?


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.


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.


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.


Reference Angle Examples

The unit circle with reference angles

The Unit Circle from Every Angle


7.3.1 Study: Terminal Conditions Study Guide

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 = _______________


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°


Page 7:




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:




120°

150°

225°

300°

330°


7.3.1 Study: Terminal Conditions Study Guide

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 = _______________


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 = __________


Pages 5 – 6:



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:





30°

2

45°

1

1

60°

2

Page 8:

Give the reference angle for each of the following angles.

a.

b.


c.

d.

e.

f.

g.

h.

i.


Pages 9 – 10:





120°

-2

135°

-1

-1

150°

2

210°

-2

225°

1

1

240°

-2

300°

2

315°

-1

-1

330°

-2


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.


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 =

_____.

# Choice

A.

B.

C.

D.


Question 5a of 10

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

# Choice

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:

# Choice

A. sin =

B. csc =

C. cos =

D. cot =


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

by is in quadrant 3":

# Choice

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 = _____.

# Choice

A.

B.

C.

D. 0


Question 9a of 10

sin( ) = _____

# Choice

A.

B.

C.

D.

Question 10a of 10

cot( ) = _____

# Choice

A. 0

B. -1

C. 1

D. Undefined


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


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.


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.


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, Again

Now 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.


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

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.

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.


8.1.1 Study: What Is a Sinusoid Anyway?

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 _____.


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.


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?


8.1.1 Study: What Is a Sinusoid Anyway?

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 _____.


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.


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?


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.


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


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


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:

apex pre-calculus learning packet · pre-calculus: weeks 3-4, april 20 – may 1 pages 3 – 4:...

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