Trigonometry- why we need to relate to circles I'm a trigonometry teaching assistant this semester and have a perhaps basic question about the motivation for using the circle in the study of trigonometry. I certainly understand Pythagorean Theorem and all that (I would hope so if I'm a teaching assistant!) but am looking for more clarification on why we need the circle, not so much that we can use it. 
I'll be more specific- to create an even angle incrementation, it seems unfeasible, for example, to use the line $y = -x+1$, divide it into even increments, and then draw segments to the origin to this lattice of points, because otherwise $\tan(\pi/3)$ would equal $(2/3)/(1/3)=2$. But why mathematically can't we do this?

The reason we use the unit circle instead of your idea is that $\cos$ and $\sin$ parametrize  the unit circle very naturally in terms of our familiar coordinates. I mean, $x^2+y^2=1$ is parametrized as $x=\cos(\theta),$ $y= \sin(\theta)$. So,using the unit circle, you can just read off the values of sine and cosine from the $x,y$ coordinates of the points on the unit circle. This is a useful thing, both computationally and conceptually.
While I suppose one could--in theory--parametrize the line $y=1-x$ using the Cosine and Sine, the parametrization will be nowhere near as neat or useful as it was above. In fact, it will be very ugly, so you can no longer read the values of sine and cosine off from the coordinates as we could in the prettier example of using the unit circle.
