Finds the angles alpha, beta, and gamma, given that numerical values for a rotation matrix are known.

euler_from_rotation_matrix.py

#!/usr/bin/env python
import numpy as np
from sympy.matrices import Matrix
from sympy import symbols, atan2, sqrt

# Conversion Factors
rtd = 180./np.pi # radians to degrees

# Fixed Axis X-Y-Z Rotation Matrix
R_XYZ = Matrix([[ 0.353553390593274, -0.306186217847897, 0.883883476483184],
            [ 0.353553390593274,  0.918558653543692, 0.176776695296637],
            [-0.866025403784439,               0.25, 0.433012701892219]])

# Calculate the Euler angles that produces a rotation equivalent to R (above)
# NOTE: the answer has units of DEGREES!
alpha = rtd*atan2(R_XYZ[1,0],R_XYZ[0,0]) # rotation about Z-axis
beta  = rtd*atan2(-R_XYZ[2,0],sqrt(R_XYZ[0,0]**2+R_XYZ[1,0]**2)) # rotation about Y-axis
gamma = rtd*atan2(R_XYZ[2,1],R_XYZ[2,2]) # rotation about X-axis