Solving triangles

Once you know a trig ratio is a fixed number for a given angle, you can use it to find missing sides — or missing angles — in any right triangle.

Finding a side from an angle

Suppose angle A = 35° and the hypotenuse is 20. You want the opposite side.

sin A = opposite / hypotenuse
→ opposite = hypotenuse × sin A = 20 × sin 35° ≈ 20 × 0.5736 ≈ 11.5

The ratio becomes a multiplier. The three forms to keep handy:

Finding an angle from two sides

If you know two sides, the inverse trig functions undo the ratio and give back the angle:

• sin⁻¹(opposite / hyp) → angle A
• cos⁻¹(adjacent / hyp) → angle A
• tan⁻¹(opposite / adj) → angle A

Example: opposite = 7, adjacent = 24.
tan A = 7 / 24 = 0.2917
A = tan⁻¹(0.2917) ≈ 16.3°

(And the third angle is 90 − 16.3 = 73.7°, since the three angles of a triangle always sum to 180°.)

The full toolkit

Any right triangle is completely determined by one angle plus one side. The ratios chain together: find one side, use it to find a second, and the right angle gives you the third angle for free. The solver tool lets you try this interactively — enter angle A and any one side to see the rest.