I can't remember whether my old Hughes Owens slide rule had trig functions on it or not. It seems to me that it didn't.

Back before there were calculators, engineers would carry around a book called the CRC Handbook, published by the Chemical Rubber Company, whoever that was. That book had tables for squares, square roots, cubes, cube roots, trig functions, hyperbolic trig fuctions, etc. You would use those tables to find the values of trig functions if you knew the angle expressed in degrees or radians.

No one ever worked out trig functions by hand, although you could do it. There were Taylor series solutions developed for all the trig functions, and the computers of the time used those series approximations to generate sine, cosine, and tangent. For example, the Taylor series for exponential of x, sine of x and cosine of x a

exp(x) = 1 + x + x2/2! + x3/3! + x4/4! + ...

sin(x) = x - x3/3! + x5/5! - x7/7! + x9/9! - ...

cos(x) = 1 - x2/2! + x4/4! - x6/6! + x8/8! - ...

where x3 means x cubed, or X times X times X, and the exclamation mark means "factorial". for example: 8! = 8 X 7 X 6 X 5 X 4 X 3 X 2 X 1 = 40320

So, you can do trig functions by hand, but the way you do them lends itself better to computer calculation.