Question

Trigonometria...

Sabendo-se que x + y = Pi/4 e sen(x) = 5/13, com 0 < x < Pi/2. Calcule sen(y) e cos(y).

Answer 1

$\displaystyle y=\frac{\pi}{4}-x\\ \\ \sin y=\sin\left(\frac{\pi}{4}-x\right)\\ \\ \sin y=\sin\left(\frac{\pi}{4}\right)\cos x-\cos\left(\frac{\pi}{4}\right)\sin x\\ \\ \sin y=\frac{\sqrt{2}}{2}\cos x-\frac{\sqrt{2}}{2}\sin x\\ \\ \sin y=\frac{\sqrt{2}}{2}\left(\cos x-\sin x\right)\\ \\ \sin y=\frac{\sqrt{2}}{2}\left(\frac{12}{13}-\frac{5}{13}\right)\\ \\ \boxed{\sin y=\frac{7\sqrt{2}}{26}}$

$\boxed{\cos y = \frac{17\sqrt{2}}{26}}$