538 CHAPTER 7 Analytic Trigonometry 51. Our Menu Has Changed “Please listen carefully; our menu has changed. On your touch-tone phone, press 1 for Sales, press 2 for Returns, etc.” For each number pressed a unique sound is produced. The sound produced is the sum of two tones, given by y lt y ht sin 2 and sin 2 π π ( ) ( ) = = where l and h are the low and high frequencies (cycles per second) shown on the illustration. For example, if you touch 7, the low frequency is l 852 = cycles per second and the high frequency is h 1209 = cycles per second.The sound emitted when you touch 7 is y t t sin 2 852 sin 2 1209 π π [ ( ) ] ( ) [ ] = + 697 cycles/sec 770 cycles/sec 852 cycles/sec 941 cycles/sec 1477 cycles/sec 1336 cycles/sec 1209 cycles/sec 1 2 3 4 5 6 7 8 9 * 0 # Touch-tone phone (a) Write this sound as a product of sines and/or cosines. (b) Determine the maximum value of y. (c) Using a graphing utility, graph the sound emitted when 7 is touched. 52. Touch-tone Phones (a) Write, as a product of sines and/or cosines, the sound emitted when the # key is touched. (b) Determine the maximum value of y. (c) Graph the sound emitted when the # key is touched. 53. Moment of Inertia The moment of inertia I of an object is a measure of how easy it is to rotate the object about Applications and Extensions In Problems 47–50, solve each equation on the interval 0 2 . θ π ≤ < 47. sin 2 sin 4 0 θ θ ( ) ( ) + = 48. cos 2 cos 4 0 θ θ ( ) ( ) + = 49. cos 4 cos 6 0 θ θ ( ) ( ) − = 50. sin 4 sin 6 0 θ θ ( ) ( ) − = some fixed point. In engineering mechanics, it is sometimes necessary to compute moments of inertia with respect to a set of rotated axes. These moments are given by the equations I I I I I I I I cos sin 2 sin cos sin cos 2 sin cos u x y xy v x y xy 2 2 2 2 θ θ θ θ θ θ θ θ = + − = + + Use Product-to-Sum Formulas to show that I I I I I I 2 2 cos 2 sin 2 u x y x y xy θ θ ( ) ( ) = + + − − and I I I I I I 2 2 cos 2 sin 2 v x y x y xy θ θ ( ) ( ) = + − − + Source: Adapted from Hibbeler, Engineering Mechanics: Statics, 13th ed., Pearson © 2013. 54. Projectile Motion The range R of a projectile propelled downward from the top of an inclined plane at an angle θ to the inclined plane is given by R v g 2 sin cos cos 0 2 2 θ θ θ φ φ ( ) ( ) = − where v0 is the initial velocity of the projectile, φ is the angle the plane makes with respect to the horizontal, and g is acceleration due to gravity. (a) Show that for fixed v0 and ,φ the maximum range down the incline is given by R v g(1 sin ) . max 0 2 φ = − (b) Determine the maximum range if the projectile has an initial velocity of 50 meters/second, the angle of the plane is 35 , φ = ° and g 9.8 meters second . 2 = 55. Derive formula (3). 56. Derive formula (7). 57. Derive formula (8). 58. Derive formula (9). 59. Challenge Problem If , α β γ π + + = show that sin 2 sin 2 sin 2 4sin sin sin α β γ α β γ ( ) ( ) ( ) + + = 60. Challenge Problem If , α β γ π + + = show that tan tan tan tan tan tan α β γ α β γ + + = 61. Solve: 27 9 x x 1 5 = − + 62. For y x 5cos 4 , π ( ) = − find the amplitude, the period, and the phase shift. 63. Find the exact value of cos csc 7 5 . 1 ( ) − 64. Find the inverse function f f x x x of 3sin 5, 2 2 . 1 π π ( ) = − − ≤ ≤ − Find the range of f and the domain and range of f .1− 65. Find the exact value of tan 6 . π ( ) − Retain Your Knowledge Problems 61–70 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.
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