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
\(\frac{y}{x}\)=\(\frac{cos2cos6cos10.....cos86}{cos1cos2cos3.....cos89}\)=\(2^{44}{2}^\frac{1}{2}\frac{cos2cos6cos10......cos86}{sin2sin4.......sin88}\)=\(2^\frac{89}{2}\frac{sin4sin8.......sin88}{sin2sin4......sin88}\)=\(\frac{2^\frac{89}{2}}{cos4cos8cos12.....cos88}\)=\(\frac{(2)^\frac{89}{2}}{(\frac{1}{2})^{22}}\) 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.
Other useful links
- https://cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA