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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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    Taylor series would be a good start – 2017-01-13
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    To approximate, use well-known values and continually average them. For example, cos(135) ~ -0.7071 and cos(120) = -0.5. Averaging these gives -0.6035. 131 is closer to 135, so average this new number with cos(135) again to get -0.6553, and the actual value is cos(131) = 0.65605... – 2017-01-13
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    It is not clear whether 1) you desire to know the method used by calculators or 2) you need an efficient method for you, for hand calculation? One thing is certain : using degrees complicates the task. It is much preferable to work with radian units – 2017-01-13

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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

https://en.wikipedia.org/wiki/CORDIC

The basic principle is similar there.