Determining $\sin(15)$, $\sin(32)$, $\cos(49)$, etc. How do you in general find the trigonometric function values? I know how to find them for 30 45, and 60 using the 60-60-60 and 45-45-90 triangle but don't know for, say $\sin(15)$ or $\tan(75)$ or $\csc(50)$, etc.. I tried looking for how to do it but neither my textbook or any other place has a tutorial for it. I want to know how to find the exact values for all the trigonometric functions like $\sin x$, $\csc x$, ... opposed to looking it up or using calculator. According to my textbook, $\sin(15)=0.26$, $\tan(75)=3.73$, and $\csc(50)=1.31$ but doesn't show where those numbers came from, as if it was dropped from the Math heaven!

The chords of the rational angles solve a series of equations, which one can derive from an iso-series, in the form  $T(n+1)=X.T(n)-T(n-1)$.  You then solve for the unique factor in each even number, and the chords of a $p$-gon solves this.   The process can be greatly accelerated, by using a bignum environment.
In any case, the exact expression of something like $cos(1°)$ solves some equation involving cube and fifth roots.  But you can get around things like this, too.
If one solves the value for a supplement chord, say $scho{15°} = (\sqrt{6}+\sqrt{2})/2) = 1.93185165259$, the chords for subsequent angles, follow the same isoseries formula as above, with X = chord 1, $T(0) = scho(0) = 2$, and $X = T(1)=scho(15°)$, and subsequent $T(n) = scho(15n°)$.  
