Trigonometry & natural numbers | PRMO 2017 | Question 11

Let f(x) =\(sin\frac{x}{3}+cos\frac{3x}{10}\) for all real x, find the least natural number x such that \(f(n\pi+x)=f(x)\) for all real x.

here f(x) =\(sin\frac{x}{3}+cos\frac{3x}{10}\)

period of\(sin\frac{x}{3}\) is \(6\pi\)

period of \(cos\frac{3x}{10}\) is \(\frac{20\pi}{3}\)

Lcm=\(\frac{60\pi}{3}\) \(\Rightarrow n=60\).

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