Pre-Calc Exam Notes 115

# Pre-Calc Exam Notes 115 - = cos 6 x sin 4 x What about the...

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Properties of Graphs of Trigonometric Functions Section 5.2 115 In general, a combination of sines and cosines will have a period equal to the lowest com- mon multiple of the periods of the sines and cosines being added. In Example 5.9, sin x and cos x each have period 2 π , so the lowest common multiple (which is always an integer multiple) is 1 · 2 π = 2 π . Example 5.10 Find the period of y = cos 6 x + sin 4 x . Solution: The period of cos 6 x is 2 π 6 = π 3 , and the period of sin 4 x is 2 π 4 = π 2 . The lowest common multiple of π 3 and π 2 is π : 1 · π 3 = π 3 1 · π 2 = π 2 2 · π 3 = 2 π 3 2 · π 2 = π 3 · π 3 = π Thus, the period of y = cos 6 x + sin 4 x is π . We can see this from its graph in Figure 5.2.9: -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 π 2 π 3 π 2 2 π y x Figure 5.2.9 y
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Unformatted text preview: = cos 6 x + sin 4 x What about the amplitude? Unfortunately we can not use the technique from Example 5.9, since we are not taking the cosine and sine of the same angle; we are taking the cosine of 6 x but the sine of 4 x . In this case, it appears from the graph that the maximum is close to 2 and the minimum is close to − 2. In Chapter 6, we will describe how to use a numerical computation program to show that the maximum and minimum are ± 1.90596111871578, respectively (accurate to within ≈ 2.2204 × 10 − 16 ). Hence, the amplitude is 1.90596111871578....
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