How does one solve for sine, cosine, or tangent (and inverse sine, cosine, tangent) without the use of a calculatour? For example: cosine of 131 degrees — how must one calculate this?
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$\begingroup$Let's figure out what happen to the earlier people who need to compute the trigonometric function out of paper and pencil or perhaps an abacus.
As a starter, we can always compute the square root using the Newton's algorithm.
Next suppose we wanted to compute $ \cos(a + b) $ such that we know $ \sin(a) $, $ \cos(a) $, $ \sin(b) $, $ \cos(b) $, we could do $ \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) $
Also, we can compute $ \cos(x) = \sqrt{\frac{1 + \cos(2x)}{2}} $
Of course, $ \sin(x) = \sqrt{1 - \cos^2(x)} $, so is also computable.
So together, given $ \sin(a) $, $ \cos(a) $, $ \sin(b) $, $ \cos(b) $, we can compute $ \sin(\frac{a + b}{2}) $ and $ \cos(\frac{a + b}{2}) $.
So given a handful of know values, we can approximate the trigonometric function by getting closer and closer to the value you want, this is basically bisection.
Of course, that is a lot of work, that is precisely why they have tables.
In a calculator, this is typically implemented using the CORDIC algorithm
The basic principle is similar there.
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