758 CHAPTER 7 Trigonometric Identities and Equations 47. cos-1 x + tan-1 x = p 2 48. sin-1 x + tan-1 x = 0 49. tan-1 x - tan-1 a 1 xb = p 6 50. tan-1 2x - tan-1 x = tan-1 1 3 The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions over the interval 30, 64. Express solutions to four decimal places. 51. 1arctanx23 - x + 2 = 0 52. psin-110.2x2 - 3 = -2x (Modeling) Solve each problem. 53. Tone Heard by a Listener When two sources located at different positions produce the same pure tone, the human ear will often hear one sound that is equal to the sum of the individual tones. Because the sources are at different locations, they will have different phase angles f. If two speakers located at different positions produce pure tones P1 = A1 sin12pƒt + f12 and P2 = A2 sin12pƒt + f22, where -p 4 … f1, f2 … p 4 , then the resulting tone heard by a listener can be written as P = Asin12pƒt + f2, where A = 21A1 cos f1 + A2 cos f222 + 1A 1 sinf1 + A2 sinf222 and f = arctana A1 sinf1 + A2 sinf2 A1 cos f1 + A2 cos f2b. (Data from Fletcher, N., and T. Rossing, The Physics of Musical Instruments, Second Edition, Springer-Verlag.) (a) Calculate A and f if A1 = 0.0012, f1 = 0.052, A2 = 0.004, and f2 = 0.61. Also, if f = 220, find an expression for P = Asin12pƒt + f2. (b) Graph Y1 = P and Y2 = P1 + P2 on the same coordinate axes over the interval 30, 0.014. Are the two graphs the same? 54. Tone Heard by a Listener Repeat Exercise 53. Use A1 =0.0025, f1 = p 7, A2 =0.001, f2 = p 6 , and f = 300. 55. Depth of Field When a large-view camera is used to take a picture of an object that is not parallel to the film, the lens board should be tilted so that the planes containing the subject, the lens board, and the film intersect in a line. This gives the best “depth of field.” See the figure. (Data from Bushaw, D., et al., A Sourcebook of Applications of School Mathematics, National Council of Teachers of Mathematics.) Subject Lens Film z y x a b (a) Write two equations, one relating a, x, and z, and the other relating b, x, y, and z. (b) Eliminate z from the equations in part (a) to obtain one equation relating a, b, x, and y. (c) Solve the equation from part (b) for a. (d) Solve the equation from part (b) for b.
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