Number of Trigonometric solutions

Algebra Level 4

For each integer $n>1$ , let $F(n)$ be the number of solutions to the equation $\sin{x} = \sin{(nx)}$ on the interval $[0,\pi]$ . What is $\displaystyle\sum_{n=2}^{2007}$ $F(n)$ ?

The answer is 2016532.

1 solution

Anuj Shikarkhane
Nov 10, 2014

$F(2) = 3$

By looking at various graphs, we obtain that, for most of the graphs

$F(n) = n + 1$

However, when $n\equiv1\pmod{4}$ , the middle apex of the sine curve touches the sine curve at the top only one time (instead of two), so we get here $F(n) = n$ . $3+4+5+5+7+8+9+9+\cdots+2008$ = $(1+2+3+4+5+\cdots+2008) - 3 - 501 = \dfrac{(2008)(2009)}{2} - 504 = 2016532$

Number of Trigonometric solutions

The answer is 2016532.

1 solution

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