Editing Euler angles (section)

===Tait–Bryan angles===
[[File:Projections_of_Tait-Bryan_angles.svg|thumb|right|Projections of ''x''-axis after three Tait–Bryan rotations. Notice that theta is a negative rotation around the axis ''y''′.]]

Assuming a frame with [[unit vector]]s (''X'', ''Y'', ''Z'') given by their coordinates as in this new diagram (notice that the angle theta is negative), it can be seen that:

:<math>\sin (\theta) = -X_3</math>

As before,
:<math>\cos^2 x = 1 - \sin^2 x,</math>
for <math> -\pi/2<x<\pi/2 </math> we have
:<math>\cos (\theta) = \sqrt {1 - X_3^2}.</math>

in a way analogous to the former one:

:<math>\sin(\psi) = X_2 / \sqrt{1 - X_3^2}.</math>
:<math>\sin(\phi) = Y_3 / \sqrt{1 - X_3^2}.</math>

Looking for similar expressions to the former ones:

:<math>\psi = \arcsin\left(X_2 / \sqrt{1 - X_3^2}\right),</math>
:<math>\theta = \arcsin(-X_3),</math>
:<math>\phi = \arcsin\left(Y_3 / \sqrt{1 - X_3^2}\right).</math>