Editing Euler angles (section)

===Proper Euler angles===
Assuming a frame with [[unit vector]]s (''X'', ''Y'', ''Z'') given by their coordinates as in the main diagram, it can be seen that:

:<math>\cos (\beta) = Z_3.</math>

And, since
:<math>\sin^2 x = 1 - \cos^2 x,</math>
for <math> 0<x<\pi </math> we have
:<math>\sin (\beta) = \sqrt {1 - Z_3^2}.</math>

As <math>Z_2</math> is the double projection of a unitary vector,
:<math>\cos(\alpha) \cdot \sin(\beta) = -Z_2,</math>
:<math>\cos(\alpha) = -Z_2 / \sqrt{1 - Z_3^2}.</math>

There is a similar construction for <math>Y_3</math>, projecting it first over the plane defined by the axis ''z'' and the line of nodes. As the angle between the planes is <math>\pi/2 - \beta</math> and <math>\cos(\pi/2 - \beta) = \sin(\beta)</math>, this leads to:

:<math>\sin(\beta) \cdot \cos(\gamma) = Y_3,</math>
:<math>\cos(\gamma) = Y_3 / \sqrt{1 - Z_3^2},</math>

and finally, using the [[inverse trigonometric functions|inverse cosine]] function,

:<math>\alpha = \arccos\left(-Z_2 / \sqrt{1 - Z_3^2}\right),</math>
:<math>\beta = \arccos\left(Z_3\right),</math>
:<math>\gamma = \arccos\left(Y_3 / \sqrt{1 - Z_3^2}\right).</math>