Question

Prove que sen(30° + α) + sen(30° – α) = cos α​

Answer 1

Sabe-se que, para dois ângulos $a$ e $b$, $\sin(a+b)=\sin a\cos b+\sin b\cos a$, logo:

$\sin(30^\circ+\alpha)=\sin30^\circ\cos\alpha+\cos30^\circ\sin\alpha$

De forma análoga, $\sin(a-b)=\sin a\cos b-\sin b\cos a$, logo:

$\sin(30^\circ-\alpha)=\sin30^\circ\cos\alpha-\cos30^\circ\sin\alpha$

Concluindo assim que:

$\sin(30^\circ+\alpha)+\sin(30^\circ-\alpha)=(\sin30^\circ\cos\alpha+\sin30^\circ\cos\alpha)+(\cos30^\circ\sin\alpha-\cos30^\circ\sin\alpha)$

$\sin(30^\circ+\alpha)+\sin(30^\circ-\alpha)=2\sin30^\circ\cos\alpha$

$\sin(30^\circ+\alpha)+\sin(30^\circ-\alpha)=2\cdot\frac{1}{2}\cdot\cos\alpha$

$\sin(30^\circ+\alpha)+\sin(30^\circ-\alpha)=\cos\alpha$