sin 10+sin 20+sin 30+sin 40+∙∙∙∙+sin 360=?


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Question Number 160762 by mathlove last updated on 06/Dec/21

$\sin 10 + \sin 20 + \sin 30 + \sin 40 + \cdot \cdot \cdot \cdot + \sin 360 = ?$
Commented by som(math1967) last updated on 06/Dec/21

$0$
Commented by mathlove last updated on 06/Dec/21

$H o w s o l u t i o n$
Commented by som(math1967) last updated on 06/Dec/21

$s i n x + s i n 2 x + . . . + s i n n x = \frac{s i n (\frac{n + 1}{2}) x s i n \frac{n x}{2}}{s i n \frac{x}{2}}$ $i f x = 10$ $∴ 10, 20, 30 . . . 360 36 t e r m$ $∴ n = 36$ $∴ s i n 10 + s i n 20 + . . . + s i n 360$ $= \frac{s i n \frac{370}{2} s i n \frac{360}{2}}{s i n \frac{10}{2}}$ $= \frac{s i n 185 s i n 180}{s i n 5} = 0$ $[∵ s i n 180 = 0]$
Commented by MJS_new last updated on 06/Dec/21

$\sin A ° = - \sin (180 + A) °$ $we {don}^{'} t need to know more$
Commented by som(math1967) last updated on 06/Dec/21

$y e s s i r$ $s i n 10 + s i n 20 + . . . + s i n 180$ $+ s i n 190 + s i n 200 + . . . s i n 360$ $= s i n 10 + s i n 20 + . . . + s i n 180$ $- s i n 10 - s i n 20 . . . + s i n 360$ $= 0$
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