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2017 AMC 12B Problem 7

Problem 7 of 25EasierAlgebraGeometry

The functions sin⁡(x)\sin(x) and cos⁡(x)\cos(x) are periodic with least period 2π.2\pi. What is the least period of the function cos⁡(sin⁡(x))?\cos(\sin(x))?

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Solution

Since cos⁡(sin⁡(x+π))\cos(\sin(x + \pi)) =cos⁡(−sin⁡(x))= \cos(-\sin(x)) =cos⁡(sin⁡(x)),= \cos(\sin(x)), the function has period π.\pi. It cannot be smaller: cos⁡(sin⁡(x))=1\cos(\sin(x)) = 1 exactly when sin⁡(x)=0,\sin(x) = 0, which happens only at integer multiples of π,\pi, so the maxima are spaced π\pi apart. The least period is π.\pi. Thus, the correct answer is B.

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Concepts: trigonometric identity · function

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