I have read this answer [Is it possible to all find trigonometric values without calculator? about how to calculate every angle without a calculator but I find it quite a bit confusing and I am wondering if there are any other methods to calculate by hand the trigonometric values of every angle.
Thanks.
It is hard to answer which angle has computable trigonometric functions.
Well we know about famous angles $0,30,45,60,90$ in degrees and their integer multiples.
Then using addition and subtraction of angles, we can calculate exact values of trigonometric functions of $15^\circ$ and $75^\circ$, as well as $22.5^\circ=45^\circ/2$.
We know exact value of trigonometric functions of angles $18, 36,54,72$ in degrees, using a pentagon, or other methods. For insance $$\sin(18^\circ)=\frac{\sqrt{5}-1}{4}. $$ Then we can compute exact value of $\sin(3^\circ)=\sin(18^\circ-15^\circ)$.
But note that $3^\circ$ is the least integer angle (in degrees) which its trigonometric functions are expressed in terms of finite radicals. This shows that any angle of the form $(3k)^\circ$ is constructible using compass and straightedge.
Note that, $\sin(1^\circ)$ is computible using the identity $$\sin(3^\circ)=3\sin(1^\circ)-4\sin^3(1^\circ) $$ but in this case, it is impossible to cancel the imaginary unit $i=\sqrt{-1}$ from inside of radicals.
About trigonometric functions of angles of the form $\frac{2\pi}{N}$, for $N=1,2,3,4,\ldots$.
Among these, for $N\leq20$, the angles $$\frac{2\pi}{7},\frac{2\pi}{9},\frac{2\pi}{11},\frac{2\pi}{13},\frac{2\pi}{14},\frac{2\pi}{18},\frac{2\pi}{19},\frac{2\pi}{20} $$ have no exact value of trigonometric functions in real radicals but it is astounding that the angle $\frac{2\pi}{17}$ is computible. In fact Gause did it. $$\cos(\frac{2\pi}{17})=-\frac{1}{16}+\frac{1}{16}\sqrt{17}+\frac{1}{16}\sqrt{\alpha}+\frac{1}{8}\sqrt{17+3\sqrt{17}-\sqrt{\alpha}-2\sqrt{\bar{\alpha}}} $$ where $$\alpha=34-2\sqrt{17}, $$ $$\bar{\alpha}=34+2\sqrt{17} $$
There is not a general simple way for that.
What we can do by hand is to start from the well known cases for $\theta =0°,30°,45°,60°,90°$ and use trigonometric identities and algebraic or geometric tricks to derive the values for trigonometric functions for other angles.
See also this related example here How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$?.
For very small angles, the Taylor development can do. But it will take several multiplies.
For arbitrary angles, I guess that the CORDIC method can be the most efficient by hand, though it requires a precomputed table of numbers. But the computations are just additions/subtractions and divisions by $2$, and a final multiply. If I am right, the amount of work will grow quadratically with the number of significant digits desired.
