2017 AMC 12B Problems/Problem 7

Problem 7

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

$\textbf{(A)}\ \frac{\pi}{2}\qquad\textbf{(B)}\ \pi\qquad\textbf{(C)}\ 2\pi \qquad\textbf{(D)}\ 4\pi \qquad\textbf{(E)}$ The function is not periodic.

Solution

$\sin(x)$ has values $0, 1, 0, -1$ at its peaks and x-intercepts. Increase them to $0, \pi/2, 0, -\pi/2$ . Then we plug them into $\cos(x)$ . $\cos(0)=1, \cos(\pi/2)=0, \cos(0)=1,$ and $\cos(-\pi/2)=0$ . So, $\cos(\sin(x))$ is $\frac{2\pi}{2} = \pi \boxed{\textbf{(B)}}$

Solution by TheUltimate123 (Eric Shen)

2017 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
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2017 AMC 12B Problems/Problem 7

Problem 7

Solution

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