Trignometric Horrors II

Geometry Level 3

Let $\beta$ = $\displaystyle \sum _{ r=1 }^{ 10 }{ \sum _{ s=1 }^{ 10 }{ \tan^{-1} { \frac { r }{ s } } } }$ .

Also $\beta$ = $\pi \gamma$ .

Then Find $\gamma$ .

Where $\tan^{-1}(x)$ Denotes Inverse Trignometric Function.

1 solution

Pratik Shastri
Nov 24, 2014

The total number of terms is $100$ .

And since $\arctan (x)+\arctan \left(\dfrac{1}{x}\right)=\dfrac{\pi}{2}$ , a pair of terms will be combined to form one $\pi/2$ .

$\therefore \beta=\dfrac{100}{2} \dfrac{\pi}{2} \implies \boxed{\gamma=25}$

Note : We need not worry about the $\arctan 1$ 's.