all correct, there is not a nice work around for general computation of trigonometric functions. this is partly why the algebra of trigonometry is emphasized. the closest one might find to simple reasonable approximations of trig functions are the approximations one uses in physics for “small angles” which is a trick employed to get a model out of a differential equation without a solution in terms of elementary functions. so for “small angles” one can substitute x for sin x and 1 for cos x.