# Finding All Solutions to Trigonometric Equations: how does one know the angles for which a given function equals some x?

For example, find all of the solutions to the equation: $$\cos\theta = \frac{1}{2}.$$

How is one to know the two angles there are for which $\cos\theta$ is equal to $\frac{1}{2}$ on the interval $[0, 2\pi)$?

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It's been a week and you've gotten two good answers. Maybe you should consider accepting one of the answers by clicking on the check mark to the left of your question. You gain some reputation and so does the person whose answer you accepted. –  Rick Decker Oct 11 '12 at 15:21

For almost all numbers like yours, the calculator's $\cos^{-1}$ button, with the calculator in radian mode, will do half the job. And for "generic" numbers, one really can't do better.

But for the special number $\frac{1}{2}$, you probably should rely on memory about special angles: there aren't many. The answer, in degrees, is $60$, so the number, in radians, is $\frac{\pi}{3}$.

For the other one, it is useful to remember the shape of the cosine curve. It is symmetric about $\pi$. So the other number is $2\pi-\frac{\pi}{3}$.

I prefer to remember the symmetry about $0$. We have $\cos(-\theta)=\cos(\theta)$, so $\cos(2\pi-\theta)=\cos(-\theta)=\cos(\theta)$.

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It's a good idea to think of trigonometric functions in terms of the unit circle. If you measure a distance $\theta$ around the unit circle, starting at the point $(1,0)$ and heading anticlockwise, you reach the point $(\cos\theta,\sin\theta)$. Which means that knowing all the $\theta$ values for which $\cos\theta=\frac 12$ is the same as knowing all the points on the unit circle where $x=\frac 12$; except that the latter is probably easier to visualise.

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