The value of sin 50° – sin 70° + sin 10°

Question:

The value of sin 50° – sin 70° + sin 10° is ___________.

Solution:

sin 50° – sin 70° + sin 10°

$=2 \sin \left(\frac{50^{\circ}-70^{\circ}}{2}\right) \cos \left(\frac{50^{\circ}+70^{\circ}}{2}\right)+\sin 10^{\circ}$

using identity $\sin a-\sin b=2 \cos \left(\frac{a+b}{2}\right) \sin \left(\frac{a-b}{2}\right)$

$=2 \sin \left(-10^{\circ}\right) \cos \left(\frac{120^{\circ}}{2}\right)+\sin 10^{\circ}$

 

$=-2 \sin 10^{\circ} \cos 60^{\circ}+\sin 10^{\circ}$         $(\because \sin (-\theta)=-\sin \theta)$

$=-2 \sin 10^{\circ}\left(\frac{1}{2}\right)+\sin 10^{\circ}$

$=-\sin 10^{\circ}+\sin 10^{\circ}$

 

$=0$

$\therefore$ Value of $\sin 50^{\circ}-\sin 70^{\circ}+\sin 10^{\circ}$ is 0

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