Solution: Cordic See page 2.

Java example 1: http://www.cl.cam.ac.uk/teaching/1314/NumMethods/JPrograms/Cordic.java

Java example 2: https://szakuljarzuk.wordpress.com/2015/12/20/problem-how-to-calculate-sin-cos-using-cordic-algorithm/

Also that it can be used for multiplication, it makes Taylor's faster too.