For simple expressions, use trigonometric/complex identities to reduce to terms you already know. For more complicated expressions, use series, ie., Taylor or Euler's expansions.
How can I work out exact values for trig functions like sin(30) and cos(3pi/2)?
Like the decimal values? Without a calculator?
I would just draw out the triangle and leave the answer in the exact radical form, like in your first example. But [itex]\cos{\frac{3\pi}{2}}[/itex] is 0 so that should be easy.