Advanced Algebra with Trig Weeks 5-6 (May 4 May 15) · 2020-04-22 · Complete Study: 11.3.1 Trig...

Advanced Algebra with Trig: Grades 11-12

CHARLES COUNTY PUBLIC SCHOOLS

Advanced Algebra with Trig Mathematics

APEX Learning Packet

Weeks 5-6 (May 4 – May 15)


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

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

Note: This content will be assessed on the Quiz at the end of the packet: Trigonometric Functions and the Unit Circle

Week 6:

Read through the Instruction and examples on the 11.3.2 The Pythagorean Theorem while completing the corresponding questions on the 11.3.2 The Pythagorean Theorem

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


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.

Advanced Algebra with Trig: Grades 11-12 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.

Advanced Algebra with Trig: Grades 11-12 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.

Advanced Algebra with Trig: Grades 11-12 Reference Angle Examples

The unit circle with reference angles

The Unit Circle from Every Angle


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 = _______________

Advanced Algebra with Trig: Grades 11-12 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°

Advanced Algebra with Trig: Grades 11-12 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:

Advanced Algebra with Trig: Grades 11-12 Fill in the missing information in the table below using the trigonometric definitions derived from the

unit circle.


120°

150°

225°

300°

330°

Advanced Algebra with Trig: Grades 11-12 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 = _______________

Pages 3 – 4:

Advanced Algebra with Trig: Grades 11-12 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 = __________



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

Advanced Algebra with Trig: Grades 11-12 Page 8:

Give the reference angle for each of the following angles.

a.

b.

c.

d.

e.

f.

g.

h.

i.



unit circle.



120°

-2

135°

-1

-1

150°

2

210°

-2

225°

1

1

240°

-2

300°

2

315°

-1

-1

330°

-2


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!


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.)


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


11.3.2 Study: Pythagorean Theorem

Study Guide

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. ______________________________


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. ______________________________

; ;

Advanced Algebra with Trig: Grades 11-12 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.

Advanced Algebra with Trig: Grades 11-12 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.


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 =


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


sin( ) = _____

A.

B.

C.

D.

Question 10a of 10

cot( ) = _____

A. 0

B. -1

C. 1

D. Undefined

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Advanced Algebra with Trig Weeks 5-6 (May 4 May 15) · 2020-04-22 · Complete Study: 11.3.1 Trig...

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