If $\sin x + \sin y = 1$ and $\cos x + \cos y = 0$, solve for $x$ and $y$ 
*

*$\sin x + \sin y = 1$

*$\cos x + \cos y = 0$


Any valid pair of $(x, y)$ is fine, as the restrictions on the board in the image below are obscured.
I got the question from chapter 26 of a comic called Yamada-kun.

How can I solve this equation?

There are so many different ways to solve it the real question is which way.
What reaches out to grab me is:
$\cos x + \cos y = 0$
$\cos x = - \cos y$ which means either $y = \pi - x$ (within a period of $2\pi$) or $y = x + \pi$ (within a period of $2\pi$).
If $y = x + \pi$ then $\sin y = - \sin x$ and $\sin y + \sin x = 0 \ne 1$ which is impossible.
If $y = x - \pi$ then $\sin y = \sin x$ and $\sin y + \sin x = 2 \sin x$.  If this is so (and it's our only option) then $\sin x = 1/2$ which means $x = \{\pi/6, 5\pi/6\}$.
So $(x,y) = (\pi/6, 5\pi/6)$ or $(x,y)= (5\pi/6, \pi/6)$ (within periods of $2\pi$)
