AMC12 2017 B

AMC12 2017 B · Q7

AMC12 2017 B · Q7. It mainly tests Functions basics, Trigonometry (basic).

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

函数 $\sin(x)$ 和 $\cos(x)$ 的最小周期为 $2\pi$。函数 $\cos(\sin(x))$ 的最小周期是多少？

(A) $\pi/2$ $\pi/2$

(B) $\pi$ $\pi$

(D) $4\pi$ $4\pi$

(E) It’s not periodic. 它不是周期函数。

Answer

Correct choice: (B)

正确答案：（B）

Solution

Because $\cos(\sin(x+\pi)) = \cos(-\sin(x)) = \cos(\sin(x))$, the function is periodic with period $\pi$. Furthermore, $\cos(\sin(x)) = 1$ if and only if $\sin(x) = 0$, which occurs if and only if $x$ is a multiple of $\pi$, so the period cannot be less than $\pi$. Therefore the function $\cos(\sin(x))$ has least period $\pi$.

因为 $\cos(\sin(x+\pi)) = \cos(-\sin(x)) = \cos(\sin(x))$，该函数以 $\pi$ 为周期。而且，$\cos(\sin(x)) = 1$ 当且仅当 $\sin(x) = 0$，这当且仅当 $x$ 是 $\pi$ 的倍数，所以周期不能小于 $\pi$。因此函数 $\cos(\sin(x))$ 的最小周期为 $\pi$。

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