Memorizing the unit circle? I know a quick google brings up plenty of resources on memorization techniques for the unit circle but I thought I would get the math stack exchange's opinion. 


*

*What is the best way to memorize the radian angles and their corresponding points on the unit circle? ( Right now I am thinking of just drawing it from scratch until I've got it.)

*Is memorizing the circle the best way to be able to quickly solve trig functions? (When first learning at least)

*How would I solve a trig function that takes an angle as an argument that has not been memorized? (You can memorize the angles that are common but there's no way you could memorize every possible angle between 0 and 2pi, is there?)
 A: *

*Best is subjective. 

*Not necessarily, I found it easier to use 30,45,60 degree right triangles.

*You cannot in general without a calculator, and you would not be expected to in general. 

A: I don't know about the fastest or "best" way, but I never bothered memorizing the unit circle.  Rather, I memorized the pictures of the sine wave and cosine waves.  (Maybe I should embed a picture here...)
Once you have the images of those curves in mind, together with the knowledge of the sine and cosine values of 0, 30, 45, 60, and 90 degrees, symmetry will take care of the rest.
For instance, if I wanted to compute $\sin 120$, I notice that it's 30 degrees from 90 degrees, so that by the symmetry of the sine wave, $\sin 120 = \sin 60 = \frac{\sqrt{3}}{2}$.
Anyway, memorizing the sine and cosine values of 0, 30, 45, 60, and 90 degrees shouldn't be too much of an issue anyway: for sine, it's just $0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1$ -- which I always remember as $$\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}$$
-- and for cosine it's the reverse.
A: *

*Varies widely.

*Varies widely.

*Depends. Any time you would encounter a trig function with very unusual arguments, it probably wouldn't be in a school exam so you'd simply use a calculator/computer. For simpler arguments which you haven't memorized, you can take advantage of half-angle, double-angle, and sum identities. For example: $\text{cos}(\frac{11\pi}{12}) = \text{cos}(\frac{2\pi}{3} + \frac{\pi}{4}) = \text{cos}(\frac{2\pi}{3})\text{cos}(\frac{\pi}{4}) - \text{sin}(\frac{2\pi}{3})\text{sin}(\frac{\pi}{4})$. These identities and much more, including memorization techniques for the unit circle, can be found here.

