Sum of Sines is a Tangent?

Geometry Level 4

$\displaystyle\sum_{k=1}^{35}\sin (5k)=\tan \left ( \dfrac mn \right )$

If $m,n$ are coprime positive integers, with $\frac m n < 90$ , find $m+n$ .

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1 solution

Pi Han Goh
Mar 30, 2015

This problem is similar to a previous problem . Read Ronak Agarwal's comment.

The proof of it is immediate after reading this .

So in this case, we get $\tan (87.5^\circ) \Rightarrow m = 175, n = 2$

We can of course pair the terms as such $( \sin(5) + \sin(175) ) + ( \sin(10) + \sin(170) ) + \ldots$

and apply sum to product rule, but that's very tedious work.

Pi Han Goh - 6 years, 2 months ago