How to find the angle of a right triangle given three sides?

You can use trigonometry

well, since it is a right triangle, you know that one of the angles is 90 degrees

sin(∅)=oposite side/hypotonuse
cos(∅)=adjacent side/hyppotnuse
tan(∅)=oposite side/adjacent side

adjacent side is the side that is next to the angle that is not the hytponuse
oposite side is the side oposite the angle

inverses can help us solve this
if sin(∅)=oposite side/hypotnuse then
sin⁻¹(oposite side/hyptonuse)=∅
note: sin⁻¹ is different from , also named cosecant
and sin⁻¹ is also called arcsin
see diagram

so for angle x we can do
sin⁻¹(b/a)=x
or
cos⁻¹(c/a)=x

for angle y we can do
sin⁻¹(c/a)=x
or
cos⁻¹(b/a)=x