Unformatted text preview: 4, so that the largest their sum could be is 3 + 4 = 7. However, 3 sin x can never equal 3 for the same x that makes 4 cos x equal to 4 (why?). 3 4 5 θ Figure 5.2.8 There is a useful technique (which we will discuss further in Chapter 6) for showing that the amplitude of y = 3 sin x + 4 cos x is 5. Let θ be the angle shown in the right triangle in Figure 5.2.8. Then cos θ = 3 5 and sin θ = 4 5 . We can use this as follows: y = 3 sin x + 4 cos x = 5 ( 3 5 sin x + 4 5 cos x ) = 5(cos θ sin x + sin θ cos x ) = 5 sin ( x + θ ) (by the sine addition formula) Thus,  y = 5 sin ( x + θ ) = 5  ·  sin ( x + θ ) ≤ (5)(1) = 5, so the amplitude of y = 3 sin x + 4 cos x is 5....
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 Fall '11
 Dr.Cheun
 Calculus, Trigonometry, Addition, Inverse Functions, Inverse function

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