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.
On
You could say the cosine function is $\frac \pi 2$ behind the sine function, although I would rather say it is $\frac \pi 2$ ahead: as you increase $\theta,$ whatever value the $\sin\theta$ reaches, $\cos\theta$ reached that value $\frac \pi 2$ radians "earlier". In other words, $\cos\theta = \sin\left(\theta + \frac \pi 2\right).$
Whichever way you look at it ("ahead" or "behind"), for any predetermined value $y$ where $-1 < y < 1,$ as you let $\theta$ increase $\sin\theta$ will attain the value $y$ twice during every interval of length $2\pi$ through which you increase $\theta$ (that is, it reaches each value between $-1$ and $1$ twice for every full cycle). If all you are told is that $\sin\theta = \frac 35,$ but you are told nothing else about $\theta,$ then you do not know which of the two possible values of $\theta$ resulted in this value of the sine function. It could be either one, there is a different value of $\cos\theta$ in each case.
$(\cos \theta,\sin \theta)$ 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.