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.
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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 $$\alpha=34-2\sqrt{17}, $$ $$\bar{\alpha}=34+2\sqrt{17} $$
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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.
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 $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$?.