一道代数题

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2019年09月02日 ⁄ 综合 ⁄ 共 183字 ⁄ 字号小中大 ⁄ 评论关闭

Thanks GT to introduce me it. It is a problem from IMO.

Find all of solutions of function about x:

$cos^2(x) + cos^2(2x) + cos^2(3x)=1$

Let's transform it to: $\frac{cos(2x)}{2} + \frac{cos(4x)}{2} + cos^2(3x)=0$

Thus, we can regard 2x as 3x - x, and regard 4x as 3x + x.

So, $cos(3x)(cos(2x) + cos(3x)) = 0$ , then, $cos(3x)cos(\frac{5x}{2})cos(\frac{x}{2}) = 0$ .

It is easy to find its solutions now.

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