// maths › Right-Angle Trigonometry
Find an unknown side of a right triangle from an acute angle and one known side, choosing sine, cosine, or tangent automatically, with a diagram whose sides update live as you drag the angle.
opp = hyp·sinθ · adj = hyp·cosθ · opp = adj·tanθ
Identify the angle and the two sides involved, choose the ratio that uses exactly those sides (sine, cosine, or tangent), then rearrange so the unknown side is the subject and substitute the known values.
Use sine for the opposite and hypotenuse, cosine for the adjacent and hypotenuse, and tangent for the opposite and adjacent. Label the sides first, then the right ratio is obvious.
With an angle of 30° and a hypotenuse of 10, to find the opposite side use opposite = hypotenuse × sin θ = 10 × sin 30° = 10 × 0.5 = 5. The same angle in radians is 30° = π/6 ≈ 0.5236 rad; switch the toggle and the working re-derives unchanged.
If the unknown is on the top of the ratio you multiply; if it is on the bottom you divide. Always keep your calculator in the same angle mode — degrees or radians — as the angle you were given.
Builders work out ramp and roof lengths, engineers size cables and supports, surveyors measure inaccessible distances, and computer graphics and robotics use these ratios to position and move objects.