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. This calculator computes all six trig functions and their inverses.

The calculator works in both degrees and radians. It can solve a triangle given three pieces of information (e.g. two sides and an angle, or three sides) using the sine rule, cosine rule and angle sum property. Results include all missing sides and angles.

The basic trig ratios — SOHCAHTOA. In right-angled triangle: sin θ = opposite ÷ hypotenuse; cos θ = adjacent ÷ hypotenuse; tan θ = opposite ÷ adjacent. SOHCAHTOA mnemonic: Sin = Opp/Hyp; Cos = Adj/Hyp; Tan = Opp/Adj. Hypotenuse: side opposite the right angle (longest side). Opposite: side opposite the angle θ. Adjacent: side next to θ (not hypotenuse). Sample 3-4-5 right triangle, θ between sides 3 and 5: sin θ = 4/5 = 0.8; cos θ = 3/5 = 0.6; tan θ = 4/3 = 1.33.

Common angles to memorise. 0°: sin=0, cos=1, tan=0. 30°: sin=½, cos=√3/2≈0.866, tan=1/√3≈0.577. 45°: sin=cos=1/√2≈0.707, tan=1. 60°: sin=√3/2, cos=½, tan=√3≈1.732. 90°: sin=1, cos=0, tan=undefined. UK GCSE and A-Level expect knowledge of exact values for 0°, 30°, 45°, 60°, 90° without calculator.

Inverse trig functions — finding angles from ratios. sin⁻¹(0.5) = 30°. cos⁻¹(0.5) = 60°. tan⁻¹(1) = 45°. Calculator buttons: sin⁻¹, arcsin, or sin with 2nd function. Used when you know two sides and want to find an angle. Sample: ladder 5m long, base 3m from wall. Height up wall = √(5² − 3²) = 4m. Angle from ground: tan⁻¹(4/3) = 53.13°.

Sine rule and cosine rule (non-right triangles). Sine rule: a/sin A = b/sin B = c/sin C — use when 2 angles + 1 side known, OR 2 sides + 1 non-included angle. Cosine rule: a² = b² + c² − 2bc·cos A — use when 3 sides known (find angle), OR 2 sides + included angle (find third side). Sample boat problem: lighthouses 1km apart, bearings 30° and 60° — use sine rule to find distance from each. Both rules in UK GCSE Higher Tier and A-Level Maths.

Real-world UK trigonometry applications. Surveying: measure heights from angles (theodolite + tape measure). GPS triangulation: position calculated from satellite angles. Construction: pitched roof angles, staircase rise/run, slope gradients. Navigation: course corrections, bearings. Physics: forces resolved into components. Engineering: stress analysis, structural calculations. Astronomy: planet positions, eclipse predictions. Music: sound waves modelled as sine waves. UK GCSE Maths Higher Tier and all A-Level Maths require trigonometry — common in real engineering and architecture roles.