Try this beautiful problem from the Pre-RMO, 2017 based on Trigonometry & natural numbers.
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.
Trigonometry
Least natural number
Functions
Answer: is 60.
PRMO, 2017, Question 11
Plane Trigonometry by Loney
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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