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!
 A: Value of $\sin{x}$ with prescribed accuracy can be calculated from Taylor's representation
$$\sin{x}=\sum\limits_{n=0}^{\infty}{\dfrac{(-1)^n x^{2n+1}}{(2n+1)!}}$$ or infinite product
$$\sin{x}=x\prod\limits_{n=1}^{\infty}{\left(1-\dfrac{x^2}{\pi^2 n^2} \right)}.$$
For some partial cases numerous trigonometric identities can be used.
A: 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 $T(n) = scho(15n°)$.  
A: For some angles (such as 15 or 75 degrees), you can apply the formulas for doubling and halving the angles and those for sums and differences of angles. For example, if you know that $\sin x=0.4$, these formulas allow you to calculate $\sin 2x$ or $\sin 17x$ or $\sin \frac{11}{32}x$, ...
Unfortunately, such formulas are not powerful enough to calculate the values for arbitrary angle (e.g. $\sin 43$). For those, one needs to employ the real-analysis definition of $\sin x$ using the Taylor series which might be beyond the scope of what you have learned so far.
