## Answers to ExercisesforSection 1: Modeling with the Sine Function

 1. Modeling with the Sine Function 2. The Six Trigonometric Functions Trigonometric Functions Main Page "Real World" Page

In the following graphs, the first-named function appears in red; the second-named function in yellow.

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 15. f(x) = 1 + sin(2x) 17. f(x) = 1.5sin(4(x0.25)) 19. f(x) = 50sin((x5)/10) 50

21. (a) Maximum sales occurred when t 4.5 (during the first quarter of 1996). Minimum sales occured when t 2.2 (during the third quarter of 1995) and t 6.8 (during the third quarter of 1996). (b) Maximum quarterly revenues were \$0.561 billion; minimum quarterly revenues were \$0.349 billion. (c) maximum: 0.455 + 0.106 = 0.561; minimum: 0.455 0.106 = 0.349

23. s(t) = 7.5sin(2(t9)/12) + 87.5

25. d(t) = 5sin(2(t1.625)/13.5)+10

27. (a) u(t) = 2.5sin(2(t0.75)) + 7.5 (b) c(t) = 1.04t[2.5sin(2(t0.75)) + 7.5

29. Amplitude = 0.106, vertical offset = 0.455, phase shift = 1.16, angular frequency = 1.39, period = 4.95. In 1995 and 1996, quarterly revenue from the sale of computers at Computer City fluctuated in cycles of 4.95 quarters about a baseline of \$0.455 billion. Every cycle, quarterly revenue peaked at \$0.561 billion (\$0.106 above the baseline) and dipped to a low of \$0.349 billion. During the second quarter of 1995 (t = 1.16) quarterly revenues were was at the baseline level and on an upward cycle.

 1. Modeling with the Sine Function 2. The Six Trigonometric Functions Trigonometric Functions Main Page "Real World" Page