Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1997 based on Trigonometry and greatest integer.
Let x=\(\frac{\displaystyle\sum_{n=1}^{44}cos n}{\displaystyle\sum_{n=1}^{44}sin n}\), find greatest integer that does not exceed 100x.
Trigonometry
Greatest Integer
Algebra
Answer: is 241.
AIME I, 1997, Question 11
Plane Trigonometry by Loney
here \(\displaystyle\sum_{n=1}^{44}cosn+\displaystyle\sum_{n=1}^{44}sin n\)
=\(\displaystyle\sum_{n=1}^{44}sinn+\displaystyle\sum_{n=1}^{44}sin(90-n)\)
=\(2^\frac{1}{2}\displaystyle\sum_{n=1}^{44}cos(45-n)\)
=\(2^\frac{1}{2}\displaystyle\sum_{n=1}^{44}cosn\)
\(\displaystyle\sum_{n=1}^{44}sin n=(2^\frac{1}{2}-1)\displaystyle\sum_{n=1}^{44}cosn\)
\(\Rightarrow x=\frac{1}{2^\frac{1}{2}-1}\)
\(\Rightarrow x= 2^\frac{1}{2}+1\)
\(\Rightarrow 100x=(100)(2^\frac{1}{2}+1)\)=241.
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