CORDIC Algorithm
How does the calculator calculate the values of trigonometric functions?
The CORDIC algorithm (short for Coordinate Rotation Digital Computer) is a clever method used by computers to calculate trigonometric functions such as sine and cosine, while using very few expensive multiplications.
CORDIC works by rotating a point in cleverly chosen, progressively smaller steps. This approach replaces most multiplications with bit shifting (dividing by powers of 2 in binary), which is very fast and simple for computers. By performing many small rotations and combining their effects, the algorithm gradually produces a close enough approximation.
Because it is efficient, CORDIC is widely used in calculators, computer graphics, robotics, and signal processing, especially in devices that do not have powerful hardware.
More information:
https://en.wikipedia.org/wiki/CORDIC
Here is the derivation of the matrix transformation:
(expressing the trigonometric functions in terms of powers of 2 and square roots)