Finding Exact Values For Trigonometry Functions (Then Using those Values to Evaluate

Trigonometry functions and Solve Trigonometry Equations)

Review: Special Right Triangles

160°

30°

1

2

3

2

1

45°

45°

2

2

2

2

π / 6

π / 3

π / 4

π / 4

Find the exact values of the missing side lengths:

The “short leg” is half the

hypotenuse

The “long leg” is the short leg

multiplied by √3

The hypotenuse is any leg multiplied by √2

30°-60°-90°

45°-45°-90° OR

Isosceles Right

0

3π / 2

π / 2

Exact Coordinates on the Unit Circle

-1

-1

1

1 0,1

1,0 1,0

30°

45°

60°

150°

135°

120°

210°

225°

240°

330°

315°

300°

0°180°

90°

π / 6

270°

π / 4π / 3

0, 1

2π / 33π / 4

5π / 6

π

7π / 6

5π / 4

4π / 3 5π / 37π / 4

11π / 6 The x and y-intercepts obviously

have exact coordinates

The angles from the

special right triangles

have exact coordinates

The angles that have the

same reference

angles as the angles from

special right

triangles have exact coordinates

1

45°2

2

2

2 1/21

30°3

21/2

1

60°

3

2

Exact Coordinates on the Unit Circle

-1

-1

1

1

3 1,

2 2

1 3,

2 2

2 2,

2 2

0,1

3 1,

2 2

1 3,

2 2

2 2

,2 2

1,0

3 1,

2 2

1 3,

2 2

2 2,

2 2

0, 1

3 1,

2 2

1 3,

2 2

2 2,

2 2

1,0

π / 6

π / 4π / 3

π / 22π / 3

3π / 4

5π / 6

7π / 6

5π / 4

4π / 33π / 2

5π / 37π / 4

11π / 6

0π

These coordinates

tell you the exact values of

cosine and sine for 16

angles.

They need to be

memorized.

Exact Coordinates on the Unit Circle

-1

-1

1

1

3 1,

2 2

1 3,

2 2

2 2,

2 2

0,1

3 1,

2 2

1 3,

2 2

2 2

,2 2

1,0

3 1,

2 2

1 3,

2 2

2 2,

2 2

0, 1

3 1,

2 2

1 3,

2 2

2 2,

2 2

1,0

30°

45°

60°

150°

135°

120°

210°

225°

240°

330°

315°

300°

0°180°

90°

270°

These coordinates

tell you the exact values of

cosine and sine for 16

angles.

They need to be

memorized.

NOTE

The coordinates on that graph tell you the exact values of cosine and sine for 16 angles. They need to be memorized for all of the included angles.

If you do not wish to memorize the unit circle or use special right triangles, the following is a trick to assist in memorization.

Reference AngleOn the left are 3 reference angles that we know exact trig values

for. To find the reference angle for angles not in the 1st quadrant (the angles at right), ignore the integer in the numerator.

0:6

3

5:4

4

0:3

6

5 7 11, ,

6 6 6

3 5 7, ,

4 4 4

2 4 5, ,

3 3 3

NOTE: Multiply the number in

the numerator

by the degree to find the angle’s

quadrant.

ExampleFind the reference angle and quadrant of the

following:34

Reference Angle:

3 45

Second Quadrant

4

Quadrant of Angle:

Or 45º

135

Stewart’s Table: Finding Exact Values of Trig Functions

R.A. Sin Cos Tan

0

6

4

3

2

0

2

1

2

2

23

2

4

2

13

22

21

2

0

1. Find the value of the Reference Angle.

2. Find the angles quadrant to figure out the sign (+/-).

0

1

2

2

21

Each time the square root number goes up by 1

Reverse the order of the values from sine

How to Remember which Trigonometric Function is Positive

-1

-1

1

1

S A

T C

AllJust Sine

Just Tangent

Just Cosine

ALLSTUDENTS

TAKE CALCULUS

Example 1

Find the exact value of the following:

34cos

Reference Angle:

Cosine of Reference Angle:

3 45 135

Sign of Cosine in Second Quadrant:

Second Quadrant

4

4cos

Quadrant of Angle:

22

AS

T C

Negative

Therefore: 234 2cos

Tho

ught

pro

cess

The only thing required for a correct answer (unless the question says explain)

Example 2Find the exact solutions to the equation below if 0 ≤ x ≤ 2π:

2cos 1x 1

2cos x Isolate the Trig

Function

2 43 3,x

1 12cosx

2.094x 120or

-1

-1

1

1120°

Are there more answers?

60°

60°

180°+60° =240°

The answer must be in Radians

Find the answer in degrees first

Find the Reference Angle

Use the reference angle to find where Cosine is also

negative

Convert the answers to radians

120

180

x

240

180

x

2

3x

4

3x

Example 3

Find the exact value of the following:

53tan

Reference Angle:

Tangent of Reference Angle:

5 60 300

Sign of Tangent in Fourth Quadrant:

Fourth Quadrant

3

3tan

Quadrant of Angle:

3

, ,

, ,

negativepositive Negative

Therefore: 53tan 3

Tho

ught

pro

cess

The only thing required for a correct answer (unless the question says explain)