1. three functions

what is for z?

this might noght be descriptive enough, so feel free to ask for clarification

thank you

2. I'm guessing you mean for the Cartesian co-ordinate system, what the trigonometric functions are.

Sine is for the y-axis, and Cosine is for the x-axis. You have it the wrong way around.

3. You can think of the $\displaystyle x,y$ plane as the "top" of a cylinder. The important things on the top can be defined by $\displaystyle (x,y)$ co-ordinates, or $\displaystyle (r, \theta)$ coordinates, where $\displaystyle x = r\cos{\theta}$ and $\displaystyle y = r\sin{\theta}$.

The height of the cylinder is defined by the length on the $\displaystyle z$ axis in $\displaystyle 3D$ space. Since this doesn't depend on $\displaystyle x$ or $\displaystyle y$, it doesn't depend on $\displaystyle r$ or $\displaystyle \theta$ either. Therefore, $\displaystyle z$ is just $\displaystyle z$.

$\displaystyle x = r\cos{\theta}$

$\displaystyle y = r\sin{\theta}$

$\displaystyle z = z$.

These are known as Cylindrical Polar Co-ordinates.

4. Or it could be the Spherical coordinate system, which if you draw a line 1 unit and move it about the origin, then find where the furthest point is from the origin.

If this is the case, then x is no longer just cosine and y is no longer just sine, because as you move it about (0,0,0) through the z-axis as well, then it will be shorter than just sin(x) and cos(x).

$\displaystyle \displaystyle{x=r*sin(\theta)*cos(\phi)}$

$\displaystyle \displaystyle{y=r*sin(\theta)*sin(\phi)}$

$\displaystyle \displaystyle{z=r*cos(\theta)}$