Finding the Exact Value of Trigonometric Functions

Finding the Exact Value of Trigonometric Functions

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 missing side Lengths:

0

3π / 2

π / 2

Important Points on 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

1

45°2

2

2

2 1/21

30°3

21/2

1

60°

3

2

Important Points on 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

Important Points on 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°

Important Points on 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

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

Then multiply the number in

the numerator

by the degree to find the angle’s

quadrant.

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

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 Reference Angle:

22

, ,

, ,

Negative

Therefore: 234 2cos

Example 2

Solve: 2sin 1 0x

1 12 2Reference Angle that Makes sin = = True:x

76 6 Solutions in the interval 0 2 :x

6

116 62

All solutions:76

116

2sin 1x 1

2sin x

2 where is an integer

2

nn

n

π / 6π / 6

-1

Slope on the Unit Circle

-1 1

1

Ө

(cosӨ,sinӨ)

cosӨ

sinӨ

Slope =

sin

cos

tan

Opposite

Adjacent

What is the slope of the terminal side of an angle on the unit circle?

opposite

adjacent

A Definition of Tangent

There are values for which the tangent function are undefined:

sintan

cos

2 5

2,

2 n For any integer n.

32, 9

2, 72, ,...11

2,

The tangent function is defined as:

Any Θ that makes cos(Θ)=0.

In general:

Stewart’s Table: Finding Exact Values of Trig Functions

R.A. Sin Cos Tan

0

6

4

3

2

00

2

1 1

2 2

2

23

2

4 21

2 2

13

22

21

2

0

0

1

1 2

3 2

1

0

2 2

2 2

3 2

1 2

1. Find the value of the Reference Angle.

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

0

1 2

2 3

1

1

3 3

3

3 2

2 1 3

Each time the square root number goes up by 1

Reverse the order of the values from sine

sintan

cos

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)