653 6.6 Graphs of the Secant and Cosecant Functions (Modeling) Distance of a Rotating Beacon A rotating beacon is located at point A next to a long wall. The beacon is 4 m from the wall. The distance d is given by d = 4 tan 2pt, where t is time in seconds since the beacon started rotating. (When t = 0, the beacon is aimed at point R. When the beacon is aimed to the right of R, the value of d is positive; d is negative when the beacon is aimed to the left of R.) Find the value of d for each time t. Round to the nearest tenth if applicable. 53. t = 0 54. t = 0.4 55. t = 1.2 56. Why is 0.25 a meaningless value for t? R A a d 4 m Graph of the Secant Function Consider the table of selected points accompanying the graph of the secant function in Figure 54 on the next page. These points include special values from -p to p. The secant function is undefined for odd multiples of p 2 and thus, like the tangent function, has vertical asymptotes for such values. Furthermore, because sec1-x2 = sec x, Even function the graph of the secant function is symmetric with respect to the y-axis. 6.6 Graphs of the Secant and Cosecant Functions ■ Graph of the Secant Function ■ Graph of the Cosecant Function ■ Techniques for Graphing ■ Connecting Graphs with Equations ■ Addition of Ordinates Relating Concepts For individual or collaborative investigation (Exercises 57 – 62) Consider the following function from Example 5. Work these exercises in order. y = -2 - cot ax - p 4b 57. What is the least positive number for which y = cot x is undefined? 58. Let k represent the number found in Exercise 57. Set x - p 4 equal to k, and solve to find a positive number for which cot Ax - p 4 B is undefined. 59. Based on the answer in Exercise 58 and the fact that the cotangent function has period p, give the general form of the equations of the asymptotes of the graph of y = -2 - cot Ax - p 4 B . Let n represent any integer. 60. Use the capabilities of a calculator to find the x-intercept with least positive x-value of the graph of this function. Round to the nearest hundredth. 61. Use the fact that the period of this function is p to find the next positive x-intercept. Round to the nearest hundredth. 62. Give the solution set of the equation -2 - cot Ax - p 4 B = 0 over all real numbers. Let n represent any integer.
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