Trigonometry Calculator

Trigonometry deals with the relationships between angles and sides of triangles. The three primary functions, sine (sin), cosine (cos) and tangent (tan), relate the angles of a right-angled triangle to the ratios of its sides, and this tool computes all six trig functions along with their inverses. The definitions are the ones behind SOHCAHTOA: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, and tan θ = opposite ÷ adjacent. The hypotenuse is the side facing the right angle and always the longest, the opposite side faces the angle θ, and the adjacent side is the remaining one touching θ.

The calculator works in both degrees and radians, and it will solve a whole triangle when you give it three pieces of information, such as two sides and an angle or all three sides, using the sine rule, the cosine rule and the fact that the angles add to 180°. Results come back with every missing side and angle filled in. To see the ratios in action, take the familiar 3-4-5 right-angled triangle with θ between the sides of length 3 and 5. Its sine is 4/5, or 0.8, its cosine is 3/5, or 0.6, and its tangent is 4/3, roughly 1.33.

A handful of angles are meant to be known exactly rather than looked up. At 0° the sine is 0, the cosine 1 and the tangent 0. At 30° the sine is ½, the cosine √3/2 (about 0.866) and the tangent 1/√3 (about 0.577). At 45° sine and cosine are both 1/√2, about 0.707, while the tangent is 1. At 60° the values swap round from 30°, so the sine is √3/2 and the cosine ½, with a tangent of √3 (about 1.732). At 90° the sine is 1, the cosine 0 and the tangent undefined. UK GCSE and A-Level papers expect all of these without a calculator.

Inverse functions run the process backwards, turning a ratio into an angle. Feed in sin⁻¹(0.5) and you get 30°, cos⁻¹(0.5) gives 60°, and tan⁻¹(1) gives 45°. On a handheld calculator they sit behind the second-function key, labelled sin⁻¹ or arcsin. You need them whenever two sides are known and the angle is not. A ladder 5m long with its base 3m from a wall reaches √(5² − 3²) = 4m up the wall, and the angle it makes with the ground is tan⁻¹(4/3), or 53.13°.

Not every triangle has a right angle, and for the rest two further rules take over. The sine rule, a/sin A = b/sin B = c/sin C, applies when you know two angles and a side, or two sides and a non-included angle. The cosine rule, a² = b² + c² − 2bc·cos A, applies when all three sides are known and you want an angle, or when two sides and the angle between them are known and you want the third side. A classic exam question sets two lighthouses 1km apart with bearings of 30° and 60°, which the sine rule solves for the distance to each. Both rules sit in GCSE Higher Tier and A-Level Maths.

Outside the exam hall the same ratios do a great deal of quiet work. Surveyors measure heights from angles using a theodolite and a tape measure, and GPS fixes a position from satellite angles by triangulation. Builders reach for trigonometry on pitched roof angles, staircase rise and run, and slope gradients, while navigators use it for bearings and course corrections. Physicists resolve forces into components with it, engineers depend on it for stress analysis and structural calculations, astronomers use it for planet positions and eclipse predictions, and sound waves in music are modelled as sine waves.