Question:
Write the value of sin 10° + sin 20° + sin 30° + ... + sin 360°.
Solution:
$\sin 10^{\circ}+\sin 20^{\circ}+\ldots+\sin 170^{\circ}+\sin 180^{\circ}+\sin \left(360^{\circ}-170^{\circ}\right)+\sin \left(360^{\circ}-160^{\circ}\right)+\ldots$$+\sin \left(360^{\circ}-20^{\circ}\right)+\sin \left(360^{\circ}-10^{\circ}\right)+\sin 360^{\circ}$
$=\sin 10^{\circ}+\sin 20^{\circ}+\ldots+\sin 180^{\circ}-\sin 170^{\circ}-\sin 160^{\circ}-\ldots-\sin 20^{\circ}-\sin 10^{\circ}+\sin 360^{\circ}$
$\left(\because \sin \left(360^{\circ}-x\right)=-\sin x\right)$
$=\sin 180^{\circ}+\sin 360^{\circ}$
$=0+0$
$=0$