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# Trigonometry & natural numbers | PRMO 2017 | Question 11

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

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=$\frac{60\pi}{3}$ $\Rightarrow n=60$.

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