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Algebra Arithmetic Math Olympiad PRMO

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

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

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