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

Question

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.

Your conclusion $\cos^2\theta = \pm \frac{4}{5}$ is wrong. Double check. — , Oct 29 '14 at 00:08
Do you mean that you expect only one value for $\cos \theta?$ — mfl, Oct 29 '14 at 00:08
I expect only one value for $cos^2\theta$. And of course $2$ for $\cos\theta$. — , Oct 29 '14 at 00:09
The square in the conclusion was a typing error. Thanks for the edit which corrected it. — , Oct 29 '14 at 00:30

score 2 · Accepted Answer · answered Oct 29 '14 at 00:09

$(\cos \theta,\sin \theta)$ 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.

score 0 · Answer 2 · answered Oct 29 '14 at 00:32

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.

Thanks for following up on the line of thought of the question. — , Oct 29 '14 at 00:44

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

2 Answers2