592 CHAPTER 8 Applications of Trigonometric Functions ‘Are You Prepared?’ Answers 1. π = = A T 5; 2 2. 0.105 rad/sec 3. π( ) = y x 7sin 6 59. A Clock Signal A clock signal is a non-sinusoidal signal used to coordinate actions of a digital circuit. Such signals oscillate between two levels, high and low,“instantaneously” at regular intervals. The most common clock signal has the form of a square wave and can be approximated by the sum of simple harmonic sinusoidal waves, such as ( ) ( ) ( ) ( ) ( ) = + + + + + f x x x x x x 2.35 sin sin 3 3 sin 5 5 sin 7 7 sin 9 9 Use a graphing utility to graph this function for π π − ≤ ≤ x 4 4 . 60. Non-sinusoidal Waves Both the sawtooth and square waves (see Problems 58 and 59) are examples of non-sinusoidal waves. Another type of non-sinusoidal wave is illustrated by the function ( ) ( ) ( ) ( ) = + + + + f x x x x x 1.6 cos 1 9 cos 3 1 25 cos 5 1 49 cos 7 Use a graphing utility to graph the function for π π − ≤ ≤ x 5 5 . 61. Use a graphing utility to graph the sound emitted by the* key on a Touch-tone phone. See Problem 51 in Section 7.7. 62. CBL Experiment The sound from a tuning fork is collected over time. A model of the form ( ) [ ] = − y A B x C cos is fitted to the data. Find the amplitude, frequency, and period of the graph. (Activity 23, Real-World Math with the CBL System) 63. CBL Experiment Pendulum motion is analyzed to estimate simple harmonic motion. A plot is generated with the position of the pendulum over time.The graph is used to find a sinusoidal curve of the form ( ) [ ] = − + y A B x C D cos . Find the amplitude, period, and frequency. (Activity 16, Real-World Math with the CBL System) 64. Challenge Problem Beats When two sinusoidal waves travel through the same medium, a third wave is formed that is the sum of the two original waves. If the two waves have slightly different frequencies, the sum of the waves results in an interference pattern known as a beat. Musicians use this idea when tuning an instrument with the aid of a tuning fork. If the instrument and the tuning fork play the same frequency, no beat is heard. Suppose two waves given by the functions, ω( ) = y t 3cos 1 1 and ω( ) = y t 3cos 2 2 where ω ω > 1 2 pass through the same medium, and each has a maximum at = t 0 sec. (a) How long does it take the sum function = + y y y 3 1 2 to equal 0 for the first time? (b) If the periods of the two functions y1 and y2 are = T 19 1 sec and = T 20 sec, 2 respectively, find the first time the sum = + = y y y 0. 3 1 2 (c) Use the values from part (b) to graph y3 over the interval ≤ ≤ x 0 600. Do the waves appear to be in tune? Explaining Concepts 65. Use a graphing utility to graph the function ( ) = > f x x x x sin , 0. Based on the graph, what do you conjecture about the value of x x sin for x close to 0? 66. Use a graphing utility to graph y x x y x x sin , sin , 2 = = and = y x x sin 3 for > x 0. What patterns do you observe? 67. Use a graphing utility to graph = = y x x y x x 1 sin , 1 sin , 2 and = y x x 1 sin 3 for > x 0. What patterns do you observe? 68. How would you explain simple harmonic motion to a friend? How would you explain damped motion? Retain Your Knowledge Problems 69–78 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 69. The function ( ) = − − ≠ f x x x x 3 4 , 4, is one-to-one. Find its inverse function. 70. Write as a single logarithm: ( ) + − + x y x y log 3log log 7 7 7 71. Solve: ( ) ( ) + + − = x x log 1 log 2 1 72. If α α π = < < cos 4 5 , 0 2 , find the exact value of: (a) α cos 2 (b) α sin 2 (c) α tan 2 73. If ( ) = − f x x 3 5 and ( ) = + g x x 7, 2 find ( ) ( ) g f x and its domain. 74. If θ = cos 5 7 and tan θ < 0, what is the value of θ csc ? 75. The normal line to a graph at a point is the line perpendicular to the tangent line of the graph at the point. If the tangent line is = − y x 2 3 1 when ( ) = f 3 1, find an equation of the normal line. 76. Solve: ( ) ⋅ − ⋅ = x x x x x 1 ln 2 0 2 2 2 77. If ( ) h x is a function with range [ ] −5, 8 , what is the range of ( ) + h x2 3 ? 78. Solve: ( )( ) − + ≤ x x x 5 3 2 0 2
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