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 = x 2 + y 2 is the distance from (0, 0) to (x, y), the six trigonometric functions of  are defined by the following ratios.

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 = -3y = -4 P = (-3, -4)  x y Example

The bottom row shows the reciprocals of the row above. Example Cont. Solution Now that we know x, y, and r, we can find the six trigonometric functions of .

Example Let tan θ = -2/3 and cos θ > 0. Find each of the six trigonometric functions of . We have to be in Quadrant IV

x y Quadrant II Sine and cosecant positive Quadrant I All functions positive Quadrant III tangent and cotangent positive Quadrant IV cosine 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:  =360 º º = 45 º

Example Find the reference angles for:

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  ´. 135 º terminates in quadrant II with a reference angle  ´ = 180 º – 135 º = 45 º. x y 135° 45° a. sin 135° Example

Solution The function value for the reference angle is sin 45 º =  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 135 º 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: