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

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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.

Use that

\sin(x + \pi) = -\sin(x)

and

\cos

is even

To rule out a smaller period, note

\cos(\sin(x)) = 1

only when

\sin(x) = 0,

i.e. at multiples of

\pi

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.