Chapter 4: Circular Functions

Lesson 4: Trigonometric Identities

Mrs. Parziale

Do Now

• Given the triangle below. How would you find the length of the missing side?

3 inches

4 inches

? inches

Trigonometric Identities

• Pythagorean Identity.

2 2sin cos 1

Example 1:

• If , then and .

Show the Pythagorean identity holds:

2 2sin cos 1

6

3cos

2

1sin

2

Example 2:

• If , use the Pythagorean Identity to

find .

1sin

3

cos

Opposites Theorem:

• Compares and -• Rotate in OPPOSITE

(clockwise) direction

cos( ) cos sin( ) sin

tan( ) tan

-

Properties of Supplements

• (For measured in radians)

sin( ) sin cos( ) cos tan( ) tan

Properties of Complements:

• (For measured in radians)

sin cos2

cos sin2

Example 3:

• Given: , without a calculator find

and

2sin

2

sin( ) sin

Example 4:

Find the following trig values:(hint: label a unit circle with appropriate coordinates at

and use those coordinates):

30, , ,2 2

sin 0 sin2

sin

3sin

2

cos0 cos2

cos 3

cos2

tan 0 tan2

tan 3

tan2

Half-Turn Theorem:

• Think in terms of what we know about odd functions. • If (x,y) is on the graph, so too is (-x, -y) because odd

functions are rotations of 180 degrees

cos( ) cos

sin( ) sin

tan( ) tan

Example 5: Calculate the exact values of the following (use your unit table knowledge to

help you.)• a) Find the following when

(First find cos, sin, and tan of each angle)

, ,3 4 6

cos( ) ________ _______ _______

sin( ) ______ _______ _______

tan( ) ______ _______ _______

Example 6:

• Given, is measured in radians and . Evaluate the following without a calculator.

a) b)

cos .6

cos( ) sin2

Example 7:

• Given, , evaluate the following without a calculator.

a) b)

5 1sin

10 4

9sin

10

sin10

Example 7, cont.:

• Given, , evaluate the following without a calculator.

c) d)

5 1sin

10 4

11sin

10

4cos

10

Closure

• If , find .

• If , find and .

• Using the unit circle, explain why for all

1sin

3 cos

sin 0.681x sin( )x sin( )x

sin( ) sin