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?

Given your comments, one of the biggest problems with the construction you've offered is that it can't define the trig functions for angles of all real values, or even for every angle between 0 and $2\pi$, since rays from the origin hit your line at angles between $-\frac{\pi}{4}$ and $\frac{3\pi}{4}$. So we'd want to at least propose some other closed curve, say a smooth one since the trig functions are smooth, as the place where we define our functions. 
Perhaps the simplest reason that the circle is a natural place to define trigonometric functions is that angle is a measure of arc of a circle.
