Tips for understanding the unit circle I am having trouble grasping some of the concepts regarding the unit circle. I think I have the basics down but I do not have an intuitive sense of what is going on. Is memorizing the radian measurements and their corresponding points the only way to master this? What are some ways one can memorize the unit circle?
Edit for clarification: I think my trouble arises by the use of $\pi$ instead of degrees. We started graphing today and the use of numbers on the number line were again being referred to with $\pi$. Why is this?

The cool thing about radians is that they relate linear measure to angular measure. So if we go $x$ radians around a circle of radius $r$ units, then we've traveled $xr$ units of length. This comes from the formula $C = 2\pi r$. You can see that $2\pi$ is the ratio of the circumference of a whole circle to its radius and that this formula can be viewed as creating a function that maps radii to circumferences. So if we go halfway around the circle, then we've traveled only half the distance and we must have $C/2 = \frac{2\pi}{2}r = \pi r$. Again, we see that $\pi$ radians can be viewed as a function mapping radii to lengths of half circles. You want a quarter circle? Well, $C/4 = \frac{2\pi}{4}r = \frac{\pi}{2}r$ and we see that to figure out the length of a quarter circular arc, one just multiplies by $\frac{\pi}{2}$.
So I would forget all about degrees and think in terms of what part of the circle an angle sweeps out. You'll get some real number between $0$ and $1$ ($1$, $\frac{1}{2}$, and $\frac{1}{4}$ in the cases we've looked at). Multiply that number by $2\pi$ and that's how many radians the angle is.
EDIT:
And note that my restriction that the number be between $0$ and $1$ is arbitrary. What about traveling around a circle twice? Then you've gone $2C = 2 \cdot 2\pi r = 4\pi r$ times around the circle and traveled around an angle of $4\pi$ radians. Nothing at all is different.
