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Try this beautiful problem from the Pre-RMO, 2017 based on Trigonometry & natural numbers.
Trigonometry & natural numbers – PRMO 2017
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.
- is 107
- is 60
- is 840
- cannot be determined from the given information
Key Concepts
Trigonometry
Least natural number
Functions
Check the Answer
But try the problem first…
Answer: is 60.
Source
Suggested Reading
PRMO, 2017, Question 11
Plane Trigonometry by Loney
Try with Hints
First hint
here f(x) =\(sin\frac{x}{3}+cos\frac{3x}{10}\)
Second Hint
period of\(sin\frac{x}{3}\) is \(6\pi\)
period of \(cos\frac{3x}{10}\) is \(\frac{20\pi}{3}\)
Final Step
Lcm=\(\frac{60\pi}{3}\) \(\Rightarrow n=60\).
Other useful links
- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA