How would you implement this not on hardware, using the same set of operators that you have available? There are series expansions for both of these high level functions. You need to research them, find the one that best fits your use case, and implement an iterative solution. For hardware you may want to unwind the iteration into a pipeline.
For example, HDL Coder uses the Newton-Raphson algorithm, with a default of 3 iterations, when implementing the Square Root block in Simulink.
Finally, if you have a fixed input range, you can often build a lookup table that implements any function over the given range. With trig functions you can typically reduce your table size to 1/4 of a full cycle by exploiting the symmetry of the function as well.
