2021 AMC 12B Fall Problem 21

Question

For real numbers

x,

let

\begin{aligned} P(x) &= 1 + \cos(x) + i\sin(x) \\ &\quad {}- \cos(2x) - i\sin(2x) \\ &\quad {}+ \cos(3x) + i\sin(3x) \end{aligned}

where

i = \sqrt{-1}.

For how many values of

x

with

0 \le x \lt 2\pi

does

P(x) = 0?

0

1

2

3

4

Write

P(x) = 1 + e^{ix} - e^{2ix} + e^{3ix}

and require both the real and imaginary parts to vanish

The imaginary part factors as

\sin(2x)(2\cos x - 1)

Group by Euler’s formula:

P(x) = 1 + e^{ix} - e^{2ix} + e^{3ix}.

The imaginary part is

\begin{gathered} \sin x - \sin 2x + \sin 3x \\ = (\sin x + \sin 3x) - \sin 2x \\ = \sin 2x(2\cos x - 1). \end{gathered}

This vanishes when

\sin 2x = 0

(so

x = 0, \tfrac{\pi}{2}, \pi, \tfrac{3\pi}{2}

) or

\cos x = \tfrac12

(so

x = \tfrac{\pi}{3}, \tfrac{5\pi}{3}

).

Checking the real part

1 + \cos x - \cos 2x + \cos 3x

at each of these values gives

\pm 2

or

1,

never

0.

So no

x

makes

P(x) = 0.

Thus, the correct answer is A.

Problem 21 in other years