Maths Olympiad Prep

Track / Stage 5 / 181 of 400 #781 of 1964

Problem 781

AIME late
Algebra Difficulty 5.3 Prove it

Function ff satisfies the equation f(cos⁡x)=cos⁡(17x)f(\cos x)=\cos (17 x). Prove that it also satisfies the equation f(sin⁡x)=sin⁡(17x)f(\sin x)=\sin (17 x).

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

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Official solution

Solution. We have
f(sin⁡x)=f(cos⁡(π2−x)=cos⁡(17π2−17x)=cos⁡(π2−17x)=sin⁡(17x). f(\sin x)=f\left(\cos \left(\frac{\pi}{2}-x\right)=\cos \left(\frac{17 \pi}{2}-17 x\right)=\cos \left(\frac{\pi}{2}-17 x\right)=\sin (17 x) .\right.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.