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 $[0, 2\pi)$?
 A: 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.
A: 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 $\cos(2\pi-\theta)=\cos(-\theta)=\cos(\theta)$.  
