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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

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