Non-geometer friendly definition of $\cos$ and $\sin$. Let $S=\{(x,y)\in \Bbb R^2: x^2+y^2=1\}$ be the unit circle.
From here, I want to define the $\sin$ and $\cos$ functions. 
I've seen some sites saying things like 

"$\cos\theta$ is the $x$ coordinate of the point $P$ where a ray from the origin hits the unit circle, when making an angle $\theta$ with the $x$-axis."

This doesn't really satisfy me: What's an angle? After answering that, aren't there two directions for $\theta$ (i.e the upper half plane and the lower half plane)?
Could anyone help me define these trigonometric functions, purely using the unit circle, and making explicit which assumptions you're making.
E: Also, in many of these definition I've read, it's not clear to me why $\cos,\sin$ are determined for every $\theta\in \Bbb R$: Say we use definition above for $\theta\in [0,2\pi)$, and extend via periodicity. 
What part says that given whatever $\theta\in [0,2\pi)$ we can trace a line from $0$, with this angle from the $x$-axis and it will hit the unit circle?

Imagine a particle traveling counterclockwise around the unit circle at a rate of one unit per second, starting at the point $(1,0)$. Then at time $\theta$ its at point $(\cos \theta, \sin \theta)$. You can define the angle traversed as $\theta$ radians.
I recommend you learn more geometry. It's fun and will make you a better math student. Then change your stackexchange handle.
