Question

TRIGONOMETRIAAAA!! Calcule o valor de $senx+sen2x+sen3x+...+sen100x$, para $x = \frac{ \pi}{2}$

Answer 1

$x=\dfrac{\pi}{2}=\dfrac{180^{\circ}}{2}=90^{\circ}$

$\text{sen}~x=\text{sen}~90^{\circ}=1$

$\text{sen}~2x=\text{sen}~180^{\circ}=0$

$\text{sen}~3x=\text{sen}~270^{\circ}=-1$

$\text{sen}~4x=\text{sen}~360^{\circ}=0$

$\text{sen}~5x=\text{sen}~450^{\circ}=1$

E assim sucessivamente.

Observe que, há um padrão a cada quatro parcelas, cuja soma é $1+0-1+0=0$.

Assim:

$\text{sen}~x+\text{sen}~2x+\text{sen}~3x+\text{sen}~4x=0$

$\text{sen}~5x+\text{sen}~6x+\text{sen}~7x+\text{sen}~8x=0$

$\dots$

$\text{sen}~97x+\text{sen}~\text{98}x+\text{sen}~99x+\text{sen}~100x=0$

A resposta é $0$.