Q:

If $\sin \theta = \frac{3}{5}$, what are the possible values of $\cos\theta$?

If $\sin \theta = \frac{3}{5}$, what are the possible values of $\cos\theta$ ?
I do know that ..
$\sin^2\theta + \cos^2\theta = 1$ 
.. and you can solve that equation, resulting in 
$\cos \theta = \pm \frac{4}{5}$
.. but why? From my understanding $\cos$ spins $\frac{1}{2}\pi$ behind $\sin$. I am not reading any books just solving this by thinking about context.

A:

$(\cos \theta,\sin \theta)$ are the coordinates for the point on the unit circle that is $\theta$ radians from $(1,0)$, measured along the circle in counterclockwise direction.
So when you know $\sin\theta=\frac35$, you know you're looking for a point that is both on the unit circle and has $y$-coordinate $\frac35$. In other words, one of the intersections between the unit circle and the line $y=\frac35$. The two possible cosine values you get out are the $x$-coordinates of those two points.

