Exercise:

Aldebaran has Right Ascension 4h36m, declination +16°31'.
What are its ecliptic coordinates?

First use
sin(β) = sin(δ) cos(ε) - cos(δ) sin(ε) sin(α)
where α = 4h36m = 69.00°, δ = 16.52°, ε = 23.43°
This gives β = -5.45°.

Now use
cos(λ) cos(β) = cos(α) cos(δ)
to obtain λ = 69.81°.

So the ecliptic co-ordinates of Aldebaran are
λ = 69.81°, β = -5.45°.

Back to "the relation between ecliptic and equatorial coordinates".