# Prove It ?? Sin10°Sin50°+Sin50°Sin250°+Sin250°Sin10°=0

May 18, 2018

Not true; the expression evaluates to –0.75.

#### Explanation:

sin 10° = 0.1736
sin 50° = 0.7660
sin 250° = –0.9397

sin 10° xx sin 50°"  " = "   "0.1330
sin 50° xx sin 250° = –0.7198
sin 250° xx sin 10° = –0.1632

=> sin 10°sin 50°+sin 50°sin 250°+sin 250°sin 10°

$= 0.1330 - 0.7198 - 0.1632$
=–0.75

May 18, 2018

$\rightarrow \sin 10 \sin 50 + \sin 50 \sin 250 + \sin 250 \sin 10$

$= \frac{1}{2} \left[2 \sin 10 \sin 50 + 2 \sin 50 \sin 250 + 2 \sin 250 \sin 10\right]$

$= \frac{1}{2} \left[\cos \left(50 - 10\right) - \cos \left(50 + 10\right) + \cos \left(250 - 50\right) - \cos \left(250 + 50\right) + \cos \left(250 - 10\right) - \cos \left(250 + 10\right)\right]$

$= \frac{1}{2} \left[\cos 40 - \cos 60 + \cos 200 - \cos 300 + \cos 240 - \cos 260\right]$

$= \frac{1}{2} \left[- \cos 60 + \cos 40 + \cos \left(180 + 20\right) - \cos \left(360 - 60\right) + \cos \left(180 + 60\right) - \cos \left(180 + 80\right)\right]$

$= \frac{1}{2} \left[- \cos 60 + \cos 40 - \cos 20 - \cos 60 - \cos 60 + \cos 80\right]$

$= \frac{1}{2} \left[- 3 \cos 60 + 2 \cos \left(\frac{80 + 40}{2}\right) \cos \left(\frac{80 - 40}{2}\right) - \cos 20\right]$

$= \frac{1}{2} \left[- 3 \cos 60 + 2 \cos 60 \cos 20 + \cos 20\right]$

$= \frac{1}{2} \left[- 3 \cdot \left(\frac{1}{2}\right) + 2 \cdot \left(\frac{1}{2}\right) \cos 20 - \cos 20\right]$

$= - \frac{3}{4}$