705 7.3 Sum and Difference Identities EXAMPLE 6 Applying the Cosine Difference Identity to Voltage Common household electric current is called alternating current because the current alternates direction within the wires. The voltage V in a typical 115-volt outlet can be expressed by the function V1t2 = 163 sin vt, where v is the angular speed (in radians per second) of the rotating generator at the electrical plant and t is time in seconds. (Data from Bell, D., Fundamentals of Electric Circuits, Fourth Edition, Prentice-Hall.) (a) It is essential for electric generators to rotate at precisely 60 cycles per sec so household appliances and computers will function properly. Determine v for these electric generators. (b) Graph V in the window 30, 0.054 by 3-200, 2004. (c) Determine a value of f so that the graph of V1t2 = 163 cos1vt - f2 is the same as the graph of V1t2 = 163 sin vt. SOLUTION (a) We convert 60 cycles per sec to radians per second as follows. v = 60 cycles 1 sec # 2p radians 1 cycle = 120p radians per sec. (b) V1t2 = 163 sin vt V1t2 = 163 sin 120pt From part (a), v = 120p radians per sec. Because the amplitude of the function V1t2 is 163, an appropriate interval for the range is 3-200, 2004, as shown in the graph in Figure 5. −200 200 0 0.05 For x = t, V(t) = 163 sin 120pt Figure 5 (c) Use the even-odd identity for cosine and a cofunction identity. cos ax - p 2b = cosc - a p 2 - xb d = cos a p 2 - xb = sin x Therefore, if f = p 2 , then V1t2 = 163 cos1vt - f2 = 163 cos avt - p 2b = 163 sin vt. S Now Try Exercise 103.
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