How Cheenta works to ensure student success?
Explore the Back-Story

Trigonometry Problem | PRMO 2016 | Question 14

Join Trial or Access Free Resources

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


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

Subscribe to Cheenta at Youtube


Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
Trial
Math Olympiad Program
magic-wandrockethighlight