Trigonometric Functions of Any

Angle

4.4

Definitions of Trigonometric Functions of Any Angle

• Let is be any angle in standard position, and let P = (x, y) be a point on the terminal side of . If r = x2 + y2 is the distance from (0, 0) to (x, y), the six trigonometric functions of are defined by the following ratios.

sin y

r, cos x

r, tan y

x, x 0

csc r

y,y 0 sec

r

x, x 0 cot

x

y,y 0

Let P = (-3, -4) be a point on the terminal side of . Find each of the six trigonometric functions of .

Solution The situation is shown below. We need values for x, y, and r to evaluate all six trigonometric functions. We are given the values of x and y. Because P = (-3, -4) is a point on the terminal side of , x = -3 and y = -4. Furthermore,

r

x = -3 y = -4

P = (-3, -4)

x

y

-5

5

-5

5

Example

222 ryx

The bottom row shows the reciprocals of the row above.The bottom row shows the

reciprocals of the row above.

sin yr

45

45

, cos xr

35

35

, tan yx

4 3

43

csc r

y

5

4

5

4, sec

r

x

5

3

5

3, cot

x

y

3

4

3

4

Example Cont.

SolutionNow that we know x, y, and r, we can find the

six trigonometric functions of .

ExampleLet tan θ = -2/3 and cos θ > 0. Find each of the six trigonometric functions of .

x

y

3

2tan

222 ryx 222 )3()2( r

r13

13

133

13

3cos

We have to be in Quadrant IV

2

3cot

3

13sec

13

132

13

2sin

2

13csc

x

y

Quadrant IISine and cosecant positive

Quadrant IAll functions

positive

Quadrant IIItangent and cotangent positive

Quadrant IVcosine and

secant positive

The Signs of the Trigonometric Functions

All Students Take Calculus

Definition of a Reference Angle

• Let be a nonacute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle ´ prime formed by the terminal side or and the x-axis.

Example

a

b

a

b

P(a, b)

Find the reference angle , for the following angle: =315º

Solution:

=360º - 315º = 45º

Example

Find the reference angles for:345

135

6

5

4

11

15345360

45180225360135

66

5

6

6

6

5

44

3

4

32

4

11

Using Reference Angles to Evaluate Trigonometric Functions

• The values of a trigonometric functions of a given angle, , are the same as the values for the trigonometric functions of the reference angle, ´, except possibly for the sign. A function value of the acute angle, ´, is always positive. However, the same functions value for may be positive or negative.

A Procedure for Using Reference Angles to Evaluate Trigonometric Functions

• The value of a trigonometric function of any angle is found as follows:

• Find the associated reference angle, ´, and the function value for ´.

• Use the quadrant in which lies to prefix the appropriate sign to the function value in step 1.

Use reference angles to find the exact value of the following trigonometric functions.

Solution

a. We use our two-step procedure to find sin 135°.Step 1 Find the reference angle, ´, and sin ´.

135º terminates in quadrant II with a reference angle ´ = 180º – 135º = 45º.

x

y

135°45°

a. sin 135°

Example

Solution

The function value for the reference angle is sin 45º = 2 / 2.

Step 2 Use the quadrant in which è lies to prefix the appropriate sign to the function value in step 1. The angle 135º lies in quadrant II. Because the sine is positive in quadrant II, we put a + sign before the function value of the reference angle. Thus, sin135= +sin45=2 / 2

Example cont.

Example

• Evaluate:

3

4cos

3cot

2

1

3cos

3

4cos

3

3

3

1

3tan

1

3cot

3cot