What exactly do the sin, cos, tan buttons do on a calculator?

The question is not very clear, so I will give two possible answers:

If you're asking what the functions sin, cos, and tan are for:

One very important property of trigonometric functions is they are uniquely determined for each angle. That's why they are so useful, we can ignore lengths and specific measurements and just say the sine of $30$ degrees is $\frac{1}{2}$

If your calculator is on the degree mode, it is computing the sine, cosine or tangent of an angle the size of the number you put in. Note that the sine of $390$ will be the same as that of $30$ since adding one full revolution doesn't change the angle.

However, if you want to know how a calculator finds values for the functions numerically, you might want to have a look at Taylor Series.

Basically, there's a way to write these functions as an infinite sum of powers of numbers, using methods of calculus. The calculator is just summing over the first few terms of this sequence to get an accurate approximation. For instance:

$$\sin(x) \approx x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040}$$

Edit: So really, if you press the sin, cos, or tan button and type in your number of degrees (°), the calculator will determine its approximate power of numbers. {Example above}: sin(x)≈x−x36+x5120−x75040

Here's a link to a very good article explaining the CORDIC algorithm: http://www.qc.cuny.edu/Academics/Degrees/DMNS/Faculty%20Documents/Sultan1.pdf

(originally published in the College Mathematics Journal)