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Problem 1198

AIME late

Algebra Difficulty 5.1 Find the answer HMMT February

Simplify the expression: $\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)^{6} + \left(\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right)^{6}$ using DeMoivre's Theorem.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

We apply DeMoivre's Theorem to simplify the first expression to $\left(\cos 6 \cdot \frac{2 \pi}{3}+\sin 6 \cdot \frac{2 \pi}{3}\right)=(\cos 4 \pi+\sin 4 \pi)=1+0=1$ . Similarly, we simplify the second expression to $\left(\cos 6 \cdot \frac{4 \pi}{3}+\sin 6 \cdot \frac{4 \pi}{3}\right)=(\cos 8 \pi+\sin 8 \pi)=1+0=1$ . Thus, the total sum is $1+1=\mathbf{2}$ .

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