5 Trigonometric Functions Copyright Cengage Learning All rights

5 Trigonometric Functions Copyright © Cengage Learning. All rights reserved.

5. 3 Trigonometric Functions of Any Angle Copyright © Cengage Learning. All rights reserved.

What You Should Learn • Evaluate trigonometric functions of any angle. • Use reference angles to evaluate trigonometric functions. • Evaluate trigonometric functions of real numbers. 3

Introduction 4

Introduction Following is the definition of trigonometric functions of Any Angle. 5

Introduction Because r = cannot be zero, it follows that the sine and cosine functions are defined for any real value of . When x = 0, however, the tangent and secant of are undefined. For example, the tangent of 90 is undefined. Similarly, when y = 0, the cotangent and cosecant of are undefined. 6

Example 1 – Evaluating Trigonometric Functions Let (– 3, 4) be a point on the terminal side of (see Figure 5. 30). Find the sine, cosine, and tangent of . Figure 5. 30 7

Example 1 – Solution Referring to Figure 4. 34, you can see that x = – 3, y = 4, and r= = 5. 8

Example 1 – Solution cont’d So, you have and 9

Introduction The signs of the trigonometric functions in the four quadrants can be determined easily from the definitions of the functions. For instance, because it follows that cos is positive wherever x > 0, which is in Quadrants I and IV. 10

Reference Angles 11

Reference Angles The values of the trigonometric functions of angles greater than 90 (or less than 0 ) can be determined from their values at corresponding acute angles called reference angles. 12

Reference Angles Figure 5. 33 shows the reference angles for in Quadrants II, III, and IV. Figure 5. 33 13

Example 4 – Finding Reference Angles Find the reference angle . a. = 300 b. = 2. 3 c. = – 135 Solution: a. Because 300 lies in Quadrant IV, the angle it makes with the x-axis is = 360 – 300 = 60 . Degrees Figure 5. 34 (a) 14

Example 4 – Solution cont’d b. Because 2. 3 lies between /2 1. 5708 and 3. 1416, it follows that it is in Quadrant II and its reference angle is = – 2. 3 0. 8416. Radians Figure 5. 34 (b) c. First, determine that – 135 is coterminal with 225 , which lies in Quadrant III. Figure 5. 34 (c) 15

Example 4 – Solution cont’d So, the reference angle is = 225 – 180 = 45 . Degrees 16

Trigonometric Functions of Real Numbers 17

Trigonometric Functions of Real Numbers To define a trigonometric function of a real number (rather than an angle), let represent any real number. Then imagine that the real number line is wrapped around a unit circle, as shown in Figure 5. 39 18

Trigonometric Functions of Real Numbers Note that positive numbers correspond to a counterclockwise wrapping, and negative numbers correspond to a clockwise wrapping. As the real number line is wrapped around the unit circle, each real number will correspond to a central angle (in standard position). Moreover, because the circle has a radius of 1, the arc intercepted by the angle will have (directional) length s = r = (1)(t) = t. 19

Trigonometric Functions of Real Numbers The point is that if is measured in radians, then t = . So, you can define sin t as sin t = sin(t radians). Similarly, cos t = cos(t radians), tan t = tan(t radians), and so on. 20

Example 8 – Evaluating Trigonometric Functions Evaluate f (t) = sin t for the following values. b. t = a. t = 1 Solution: a. f (1) = sin 1 0. 8415 Radian mode b. f ( ) = sin =0 Common angle 21

Trigonometric Functions of Real Numbers The domain of the sine and cosine functions is the set of all real numbers. To determine the range of these two functions, consider the unit circle shown in Figure 5. 40 22

Trigonometric Functions of Real Numbers Because r = 1, it follows that sin t = y and cos t = x. Moreover, because (x, y) is on the unit circle, you know that – 1 y – 1 and – 1 x – 1. So, the values of sine and cosine also range between – 1 and 1. – 1 y – 1 x – 1 and – 1 sin t 1 – 1 cos t 1 23

Trigonometric Functions of Real Numbers You can add 2 to each value of in the interval [0, 2 ] completing a second revolution around the unit circle, as shown in Figure 5. 41 24

Trigonometric Functions of Real Numbers The values of Sin(t + 2 ) Cos(t + 2 ) and correspond to those of sin t and cos t. Similar results can be obtained for repeated revolutions (positive or negative) on the unit circle. This leads to the general result Sin(t + 2 n) = sin t and Cos(t + 2 n) = cos t for any integer and real number Functions that behave in such a repetitive (or cyclic) manner are called periodic. 25

Trigonometric Functions of Real Numbers From this definition it follows that the sine and cosine functions are periodic and have a period of 2. The other four trigonometric functions are also periodic. A function f is even when f (–t) = f (t) and is odd when f (–t) = f (t). 26

Trigonometric Functions of Real Numbers 27