13.3 Evaluate Trigonometric Functions of Any Angle

Unit circle

The circle x² + y² = 1, which has center (0, 0) and radius 1

Quadrantal angle

An angle in standard position whose terminal side lies on an axis

Reference angle

The reference angle for q is the acute angle q’ formed by the terminal side of q and the
x-axis.

GENERAL DEFINITIONS OF TRIGONOMETRIC FUNCTIONS

Let q be an angle in standard position, and let (x, y) be the point where the terminal side of q intersects the circle x² + y² = r². The six trigonometric functions of q are defined as follows:

sin q =

csc q = ,y ¹ 0

cos q =

sec q = ,x ¹ 0

tan q = ,x ¹ 0

cot q = ,y ¹ 0

Example 1

Evaluate trigonometric functions given a point

Let (-12, 5) be a point on the terminal side of an angle q in standard position. Evaluate the six trigonometric functions of q.

Use the Pythagorean theorem to find the value of r.

r = = = = _13_

Using x = -12, y = 5, and r = _13_ , you can write:

sin q = =

csc q = =

cos q = =

sec q = =

tan q = =

cot q = =

THE UNIT CIRCLE

The circle x² + y² = 1, which has center (0, 0) and radius 1, is called the unit circle.

sin q = = =_y_

cos q = = =_x_

Example 2

Use the unit circle

Use the unit circle to evaluate the six trigonometric functions of q = 450°.

Draw the unit circle, then draw the angle q = 450° in standard position. The terminal side of q intersects the unit circle at (_0_, _1_), so use x = _0_ and y = _1_ to evaluate the trigonometric functions.

sin q = = =_1_	csc q = = =_1_
cos q = = =_0_	sec q = = _undefined_
tan q = = _undefined_	cot q = = =_0_

REFERENCE ANGLE RELATIONSHIPS

Let q be an angle in standard position. The reference angle for q is the acute angle q’ formed by the terminal side of q and the x-axis. The relationship between q and q’ is shown below for nonquadrantal angles q such that

90° < q < 360°( < q < 2p).

Quadrant II	Quadrant III	Quadrant IV
Degrees:
q’ =180° - q	q’ = q - 180°	q’ =360° - q
Radians:
q’ =p - q	q’ =q – p	q’ =2p - q

Example 3

Find reference angles

Find the reference angle q’ for (a) q = -165° and

(b) q = .

Solution

a. Note that q is coterminal with _195°_, whose terminal side lies in Quadrant _III_. So,

q’ = _195°_ - _180°_ = _15°_ .

b. The terminal side of q lies in Quadrant _IV_ . So,

q’ = _2p_- = .

EVALUATING TRIGONOMETRIC FUNCTIONS

Use these steps to evaluate a trigonometric function for any angle q:

STEP 1	Find the reference angle _q’_.
STEP 2	Evaluate the trigonometric functions for _q’_.
STEP 3	Determine the sign of the trigonometric function value from the quadrant in which _q_ lies.

Example 4

Use reference angles to evaluate functions

Evaluate (a) cos(-225° ) and (b) cot .

a. The angle -225° is coterminal with _135°_. The reference angle is q’ = _180°_ - _135°_ = _45°_ . The cosine function is negative in Quadrant _II_, so you can write:

cos(-225°) = -cos( _45°_) =

b. The angle is coterminal with . The reference

angle is q' = - _p_= . The cotangent

function is positive in Quadrant _III_, so you can write:

cot ( ) =cot =