Approximating transcendental functions is an art beginning with simple expansions involving taylor series, and then, often, quickly moving to Chebyshev polynomials and other more advanced techniques. Numerical recipies has a chapter on this topic:
- Numerical Recipes http://www.aip.de/groups/soe/local/numres/bookcpdf/c5-8.pdf
- Approximation Theory and Approximation Practice https://people.maths.ox.ac.uk/trefethen/ATAP/ATAPfirst6chapters.pdf
NASA's published approximations: