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Can you calculate arbitrary sines/cosines by using angle addition and double angle formulas? I thought that Taylor Series was the standard for calculating the sine of arbitrary angles. What is I mean is an angle like $59$ is the same as $118=90+28...$ and through successive splittings we can break our angle into angles, like $90$, we know the value of sine/cosine at.

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Tables are actually quite common for fast computing like graphics. A friend of mine had a nice idea once (he probably wasn't the first) to represent the angle in binary and use half angle and sum formulas. –  DanielV Feb 27 at 7:09
    
See exact trigonometric constants. –  Lucian Feb 27 at 7:16

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The famous CORDIC algorithm (last seen discussed here) actually implements this idea.

A more primitive version is that one knows some trig. values algebraically, with some effort for all multiples of $3°$. Also, the complex square root can be reduced to real square roots, so halving an angle is also possible.

So for example, to compute (approximate) $\sin$ and $\cos$ for $59°$ you could split it as $60°-1°$ and then use the binary expression of $1°/3°=(0.010101...)_2=\frac14+\frac1{16}+\frac1{64}+...$ to compose $\cos1°+i\sin1°$ starting with the values for $3°$ as obtained from the known exact expressions for $90°-15°-72°$, and then using fractions of it obtained by repeated angle bisection, for $3°/4=0.75°$, $3°/16=0.1875°$, $3°/64=0.046875°$ etc.

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