660 CHAPTER 6 The Circular Functions and Their Graphs Work each problem. 37. Concept Check If c is any number such that -1 6c 61, then how many solutions does the equation c = sec x have over the entire domain of the secant function? 38. Concept Check Consider the function g1x2 = -2 csc14x + p2. What is the domain of g? What is its range? 39. Show that sec1-x2 = sec x by writing sec1-x2 as 1 cos1-x2 and then using the relationship between cos1-x2 and cos x. 40. Show that csc1-x2 = -csc x by writing csc1-x2 as 1 sin1-x2 and then using the relationship between sin1-x2 and sin x. Concept Check Decide whether each statement is true or false. If false, explain why. 33. The tangent and secant functions are undefined for the same values. 34. The secant and cosecant functions are undefined for the same values. 35. The graph of y = sec x in Figure 56 suggests that sec1-x2 = sec x for all x in the domain of sec x. 36. The graph of y = csc x in Figure 59 suggests that csc1-x2 = -csc x for all x in the domain of csc x. 27. p 4 –p 8 –p 4 p 8 x y –1 1 0 28. x y –1 1 0 p 2 p 4 –p 4 –p 2 29. x y 0 3 1 –1 –3 p 2 p 3p 2p 2 30. p 2 x y 0 2 1 –1 –2 p 3p 2p 2 31. x y –2 2 0 p p 2 –p 2 –p 32. x y –2 –1 2 1 0 p 3p 2p 4p Connecting Graphs with Equations Determine an equation for each graph. See Example 3. (Modeling) Distance of a Rotating Beacon A rotating beacon is located at point A, 4 m from a wall. The distance a is given by a = 4 sec 2pt , where t is time in seconds since the beacon started rotating. Find the value of a for each time t. Round to the nearest tenth if applicable. 41. t = 0 42. t = 0.86 43. t = 1.24 44. t = 0.25 R A a d 4 m
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