746 CHAPTER 7 Trigonometric Identities and Equations (b) All angles 2u that are solutions of the equation sin 2u = 23 2 are found by adding integer multiples of 360° to the basic solution angles, 60° and 120°. 2u = 60° + 360°n and 2u = 120° + 360°n u = 30° + 180°n and u = 60° + 180°n Divide by 2. All solutions are given by the following set, where 180° represents the period of sin 2u. 530° + 180°n, 60° + 180°n, where n is any integer6 S Now Try Exercises 75 and 99. Add integer multiples of 360°. Applications EXAMPLE 9 Describing a MusicalTone from a Graph A basic component of music is a pure tone. The graph in Figure 34 models the sinusoidal pressure y = P, in pounds per square foot, from a pure tone at time x = t in seconds. −0.006 0.006 0 0.04 Figure 34 −0.006 0.006 0 0.02 y1 = 0.004 sin 300px y2 = 0.004 Figure 35 (a) The frequency of a pure tone is often measured in hertz. One hertz is equal to one cycle per second and is abbreviated Hz. What is the frequency f, in hertz, of the pure tone shown in the graph? (b) The time for the tone to produce one complete cycle is the period. Approximate the period T, in seconds, of the pure tone. (c) An equation for the graph is y = 0.004 sin 300px. Use a calculator to estimate all solutions that make y = 0.004 over the interval 30, 0.024. SOLUTION (a) From Figure 34, we see that there are 6 cycles in 0.04 sec. This is equivalent to 6 0.04 = 150 cycles per sec. The pure tone has a frequency of ƒ = 150 Hz. (b) Six periods cover a time interval of 0.04 sec. One period would be equal to T = 0.04 6 = 1 150 sec, or 0.006 sec. (c) If we reproduce the graph in Figure 34 on a calculator as y1 and also graph a second function as y2 = 0.004, we can determine that the approximate values of x at the points of intersection of the graphs over the interval 30, 0.024 are 0.0017, 0.0083, and 0.015. The first value is shown in Figure 35. These values represent time in seconds. S Now Try Exercise 105.
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