Tricky Trigometry

Geometry Level 3

$\large \dfrac{\sin 50^\circ+\sin 100^\circ+\sin 210^\circ}{(\sin 25^\circ)(\sin 50^\circ)(\sin 105^\circ)} =\, ?$

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2 solutions

Akshay Sharma
Mar 22, 2016

Let $2 A=50^o$ , $2 B=100^o$ , $2 C=210^o$

$\quad\rightarrow$ $A=25^o$ , $B=50^o$ , $C=105^o$

Since, $A+B+C=180^o$ ,We know that

$\boxed {sin 2 A+sin 2 B +sin 2 C=4(sin A)(sin B)(sin C)}$

$=$ $\frac{sin 2(25^o)+sin 2(50^o)+sin 2(105^o)}{(sin 25^o)(sin 50^o)(sin 105^o)}$

$=$ $4\frac{(sin 25^o)(sin 50^o)(sin 105^o)}{(sin 25^o)(sin 50^o)(sin 105^o)}$

$=$ $\boxed {4}$

Umair Siddiqui
Mar 22, 2016

use 2 sinA sinB = cos (A-B) - cos (A+B) in the denominator.....then convert cos to sin