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# 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

Nearest Integer

Algebra

Trigonometry

## Check the Answer

PRMO, 2016, Question 14

Plane Trigonometry by Loney

## Try with Hints

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

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

=$$\frac{2}{7} \times \frac{133}{2}$$=19.

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