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

2017 AMC 12B Problem 7 — Solution

2017 AMC 12B Problem 7

Below is the professionally curated solution for Problem 7 of the 2017 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 12B solutions, or check the answer key.

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Concepts:trigonometric identity function

Difficulty rating: 1500

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

$\dfrac{\pi}{2}$

$\pi$

$2\pi$

$4\pi$

It's not periodic.

Solution:

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

Thus, the correct answer is B.

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