Pre-Calc Exam Notes 114

Pre-Calc Exam Notes 114 - 4 so that the largest their sum...

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114 Chapter 5 Graphing and Inverse Functions §5.2 Example5.9 Find the amplitude and period of y = 3 sin x + 4 cos x . Solution: This is sometimes called a combination sinusoidal curve, since it is the sum of two such curves. The period is still simple to determine: since sin x and cos x each repeat every 2 π radians, then so does the combination 3 sin x + 4 cos x . Thus, y = 3 sin x + 4 cos x has period 2 π . We can see this in the graph, shown in Figure 5.2.7: -5 -4 -3 -2 -1 0 1 2 3 4 5 0 π 2 π 3 π 2 2 π 5 π 2 3 π 7 π 2 4 π y x Figure 5.2.7 y = 3 sin x + 4 cos x The graph suggests that the amplitude is 5, which may not be immediately obvious just by looking at how the function is defined. In fact, the definition y = 3 sin x + 4 cos x may tempt you to think that the amplitude is 7, since the largest that 3 sin x could be is 3 and the largest that 4 cos x could be is
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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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