solving right triangles
Solving a triangle means finding all its unknown sides and angles from the few you are given. For a right triangle this is the cleanest case in all of trigonometry, because the right angle and the trig ratios together pin everything down once you have just two more pieces of information.
A right triangle has three sides and three angles, and one angle is already known to be 90 degrees. So you need only two further facts to find the rest — and at least one must be a side, since angles alone fix the shape but not the size. The toolkit is small: the angle sum (the two acute angles add to 90 degrees, so each is the other's complement), the Pythagorean theorem a^2 + b^2 = c^2 for the sides, and SOHCAHTOA to connect an angle to a pair of sides. A typical recipe: given one acute angle and one side, use a trig ratio to get a second side, then Pythagoras or another ratio for the third, and 90 degrees minus the known angle for the last angle. To recover an angle from two sides you use an inverse trig function, for example theta = arctan(opposite / adjacent).
This is the engine behind surveying, navigation, and construction. Any slanted real-world distance — the height of a tree, the span of a roof, the reach of a crane — becomes a right-triangle side you can compute from an angle you can measure. Drawing and labelling the triangle before reaching for a formula prevents most mistakes.
A right triangle has hypotenuse 10 and one acute angle 35 degrees. The side opposite that angle is 10 x sin 35 degrees ≈ 5.74; the adjacent side is 10 x cos 35 degrees ≈ 8.19; and the other acute angle is 90 - 35 = 55 degrees. The triangle is now fully solved.
Two extra facts (here a side and an angle) plus the built-in right angle are exactly enough to find everything else.
You cannot 'solve' a right triangle from its three angles alone — angles fix the shape but leave the size free, so you always need at least one side.