Question

Write the value of sin 10° + sin 20° + sin 30° + ... + sin 360°.

Answer 1

\[\sin10^\circ + \sin20^\circ + . . . + \sin170^\circ + \sin180^\circ + \sin\left( 360^\circ - 170^\circ \right) + \sin\left( 360^\circ - 160^\circ \right) + \sin\left( 360^\circ - 10^\circ \right) + \sin360^\circ\]
\[ = \sin10^\circ + \sin20^\circ + . . . + \sin180^\circ - \sin170^\circ - \sin160^\circ - . . . - \sin20^\circ - \sin10^\circ + \sin360^\circ \left( \because \sin\left( 360^\circ - x \right) = - \sin x \right)\]
\[ = \sin180^\circ + \sin360^\circ\]
\[ = 0 + 0\]
\[ = 0\]