A surprisingly good recursive approximation of sin and cos using double angle theorem.

sin_cos_approx.py

from math import sqrt, pi, sin, cos

def my_sin(x):
    if x < 0.01:
        return x
    return 2 * my_sin(x/2) * my_cos(x/2)

def my_cos(x):
    if x < 0.01:
        s = my_sin(x)
        return sqrt(1 - s * s)

    s = my_sin(x / 2)
    c = my_cos(x / 2)
    return c * c - s * s

for x in range(0, 361, 30):
    angle = x * pi / 180
    mine_sin = my_sin(angle)
    actual_sin = sin(angle)
    sin_error = (mine_sin - actual_sin)/max(actual_sin, 0.01) * 100

    mine_cos = my_cos(angle)
    actual_cos = cos(angle)
    cos_error = (mine_cos - actual_cos)/max(actual_cos, 0.01) * 100
    print(f"Angle: {x}, My Sin: {mine_sin:.5f}, Actual Sin: {actual_sin:.5f}, Sin Error: {sin_error:.5f}% My Cos: {mine_cos:.5f}, Actual Cos: {actual_cos:.5f}, Cos Error: {cos_error:.5f}%")