Q:

How to find sin and cos of 0, pi/2, pi without calculator

In my notes it shows how to calculate by using the unit circle. But I do not know why the value of sin is the y coordinate and the value of cos is the x coordinate.

A:

I'm not sure what your difficulty is, so I'll use the geometrical definition to get the requested values. This definition means that when going a distance of $\theta$ counter-clockwise along the unit circle starting at $(1,0)$, the coordinates of the point on the circle are $(\cos \theta, \sin \theta)$.
Note that $0$ corresponds to the point $(1,0)$ on the unit circle, i.e.
$$\sin(0) = 0, \cos(0) = 1$$
$\frac\pi2$ represents a quarter of a circle ($90^\circ$) wich corresponds to $(0,1)$ on the unit circle, i.e.
$$\sin(\frac\pi2) = 1, \cos(\frac\pi2) = 0$$
Finally, $\pi$ corresponds to half a circle ($180^\circ$) wich corresponds to $(-1,0)$. This means
$$\sin(\pi) = 0, \cos(\pi) = -1$$

