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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

### Key Concepts

Nearest Integer

Algebra

Trigonometry

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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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

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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