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April 16, 2020

Trigonometry Problem | PRMO 2016 | Question 14

Try this beautiful problem from the PRMO, 2016 based on trigonometry and nearest integer.

Trigonometry and nearest integer - PRMO 2016


If x=cos1cos2cos3.....cos89 and y=cos2cos6cos10....cos86 then find the integer nearest to \(\frac{2}{7}log_{2}(\frac{y}{x})\)

  • is 107
  • is 19
  • is 840
  • cannot be determined from the given information

Key Concepts


Nearest Integer

Algebra

Trigonometry

Check the Answer


Answer: is 19.

PRMO, 2016, Question 14

Plane Trigonometry by Loney

Try with Hints


First hint

\(\frac{y}{x}\)=\(\frac{cos2cos6cos10.....cos86}{cos1cos2cos3.....cos89}\)=\(2^{44}{2}^\frac{1}{2}\frac{cos2cos6cos10......cos86}{sin2sin4.......sin88}\)=\(2^\frac{89}{2}\frac{sin4sin8.......sin88}{sin2sin4......sin88}\)=\(\frac{2^\frac{89}{2}}{cos4cos8cos12.....cos88}\)=\(\frac{(2)^\frac{89}{2}}{(\frac{1}{2})^{22}}\) for given condition by question =\(2^\frac{133}{2}\)

Second Hint

\(\frac{2}{7}log_{2}(\frac{y}{x})\)=\(\frac{2}{7}log_{2}{2}^\frac{133}{2}\)

Final Step

=\(\frac{2}{7} \times \frac{133}{2}\)=19.

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