Question

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

Answer 1

When you are given the sides of the right angle but not the angles you can use inverse trig. In inverse trig you can remember this to find the angles...
SOH CAH TOA

Sin= Opposite over Hypotenuse
Cos= Adjacent over Hypotenuse
Tan= Opposite over Adjacent

Hope I helped!!!

Answer 2

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 $\frac{1}{sin}$, 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