Question

sin9°+sin49°+...+sin289°+sin329°

Answer 1

Рассмотрим следующий пример:

sin( α + β ) = sinα•cosβ + cosα•sinβ

sinα + sin(120° + α) + sin(240° + α) = sinα + (sin120°•cosα + cos120°•sinα) + (sin240°•cosα + cos240°•sinα) = sinα + (√3/2)•cosα - (1/2)•sinα - (√3/2)•cosα - (1/2)•sinα = sinα - sinα = 0

Аналогично

sin9° + sin49° + sin89° + sin129° + sin169° + sin209° + sin249° + sin289° + sin329° =  (sin9° + sin129° + sin249°) + (sin49° + sin169° + sin289°) + (sin89° + sin209° + sin329°) = (sin9° + sin(120° + 9°) + sin(240° + 9° )) + (sin49° + sin(120 + 49°) + sin(240° + 49°)) + (sin89° + sin(120° + 89°) + sin(240° + 89°)) = 0 + 0 + 0 = 0

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