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310 Chapter 4 Trigonometry 4.4 TRIGONOMETRIC FUNCTIONS OF ANY ANGLE What you should learn • Evaluate trigonometric functions of any angle. • Find reference angles. • Evaluate trigonometric functions of real numbers. Why you should learn it You can use trigonometric functions to model and solve real-life problems. For instance, in Exercise 99 on page 318, you can use trigonometric functions to model the monthly normal temperatures in New York City and Fairbanks, Alaska. Introduction In Section 4.3, the definitions of trigonometric functions were restricted to acute angles. In <strong>this</strong> section, the definitions are extended to cover any angle. If is an acute angle, these definitions coincide with those given in the preceding section. Definitions of Trigonometric Functions of Any Angle Let be an angle in standard position with x, y a point on the terminal side of and r x 2 y 2 0. y sin cos x r ( x, y) tan y x , x 0 sec y r cot x y , y 0 r x , x 0 csc r y , y 0 r θ x Because r x 2 y 2 cannot be zero, it follows that the sine and cosine functions are defined for any real value of . However, if x 0, the tangent and secant of are undefined. For example, the tangent of 90 is undefined. Similarly, if y 0, the cotangent and cosecant of are undefined. Example 1 Evaluating Trigonometric Functions James Urbach/SuperStock Let 3, 4 be a point on the terminal side of . Find the sine, cosine, and tangent of . Solution Referring to Figure 4.36, you can see that x 3, y 4, and r x 2 y 2 3 2 4 2 25 5. So, you have the following. sin y r 4 5 cos x r 3 5 tan y x 4 3 y ( −3, 4) r 4 3 2 The formula r x 2 y 2 is a result of the Distance Formula. You can review the Distance Formula in Section 1.1. Now try Exercise 9. 1 θ −3 −2 −1 1 FIGURE 4.36 x
Section 4.4 Trigonometric Functions of Any Angle 311 π 2 < θ < π x < 0 y > 0 y 0 < θ < π 2 x > 0 y > 0 The signs of the trigonometric functions in the four quadrants can be determined from the definitions of the functions. For instance, because cos it follows that cos is positive wherever x > 0, which is in Quadrants I and IV. (Remember, r is always positive.) In a similar manner, you can verify the results shown in Figure 4.37. xr, x Example 2 Evaluating Trigonometric Functions x < 0 y < 0 π < θ < 3π 2 y x > 0 y < 0 3π 2 < θ < 2π Given tan and cos > 0, find sin and Solution 5 4 sec . Note that lies in Quadrant IV because that is the only quadrant in which the tangent is negative and the cosine is positive. Moreover, using Quadrant II sin θ: + cos θθ : tan : − − Quadrant III sin θ: − cos θθ : − tan : + FIGURE 4.37 Quadrant I sin θ: + cos θθ : + tan : + Quadrant IV sin θ: − cos θθ : + tan : − x tan y x 5 4 and the fact that y is negative in Quadrant IV, you can let y 5 and x 4. So, r 16 25 41 and you have sin y r 5 41 0.7809 sec r x 41 4 1.6008. Now try Exercise 23. Example 3 Trigonometric Functions of Quadrant Angles (−1, 0) FIGURE 4.38 π 2 3π 2 y (0, 1) π 0 (0, −1) (1, 0) x Evaluate the cosine and tangent functions at the four quadrant angles 0, and 2 , , 2 . Solution To begin, choose a point on the terminal side of each angle, as shown in Figure 4.38. For each of the four points, r 1, and you have the following. cos 0 x tan 0 y x, y 1, 0 x 0 r 1 1 1 1 0 cos tan 2 y x 1 2 x r 0 1 0 0 ⇒ undefined cos x tan y x, y 1, 0 x 0 r 1 1 0 3 cos tan 3 x, y 0, 1 2 y x 1 2 x r 0 1 0 0 ⇒ undefined 1 1 x, y 0, 1 Now try Exercise 37. 3
Page 1 and 2: Trigonometry 4 4.1 Radian and Degre
Page 3 and 4: Section 4.1 Radian and Degree Measu
Page 15 and 16: Section 4.2 Trigonometric Functions
Page 21 and 22: Section 4.3 Right Triangle Trigonom
Page 31: Section 4.3 Right Triangle Trigonom
Page 41 and 42: Section 4.5 Graphs of Sine and Cosi
Page 53 and 54: Section 4.6 Graphs of Other Trigono
Page 63 and 64: Section 4.7 Inverse Trigonometric F
Page 73: ERROR: rangecheck OFFENDING COMMAND

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