A right triangle is fully determined by any two of its sides, because the right angle fixes everything else. Pythagoras gives the third side and the inverse trigonometric functions give the angles.
Which ratio to reach for
The mnemonic SOH-CAH-TOA encodes it: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. When you know two sides and want an angle, pick the ratio that uses the two sides you have and apply its inverse.
The two acute angles always sum to 90°
Because the interior angles of any triangle total 180° and one of them is already 90°. This is why sin A = cos B in a right triangle — the two angles are complementary, and that identity is where the “co” in cosine comes from.
Pythagorean triples
Some right triangles have all three sides as whole numbers: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Any multiple of a triple is another triple, which is why builders use the 3-4-5 method to square a corner with nothing more than a tape measure.