Trigonometry Calculator — Sine, Cosine & Tangent

Trigonometry Guide

SOH CAH TOA

In a right triangle: sin θ = Opposite/Hypotenuse. cos θ = Adjacent/Hypotenuse. tan θ = Opposite/Adjacent. Finding a side: multiply the known side by the trig ratio. Finding an angle: use inverse trig (arcsin, arccos, arctan). Example: right triangle with hypotenuse 10, angle 30°: opposite = 10 × sin 30° = 10 × 0.5 = 5. adjacent = 10 × cos 30° = 10 × 0.866 = 8.66. Special angles: sin 30° = 0.5, sin 45° = √2/2 ≈ 0.707, sin 60° = √3/2 ≈ 0.866. These should be memorised — exact values are required i

Sine Rule

a/sin A = b/sin B = c/sin C. Used when: two angles and one side known (AAS or ASA). Two sides and a non-included angle known (SSA — ambiguous case). Finding side: a = b × sin A / sin B. Finding angle: sin A = a × sin B / b. Example: a=5, angle A=40°, angle B=70°. b = 5 × sin 70° / sin 40° = 5 × 0.940 / 0.643 = 7.31. Ambiguous case (SSA): two triangles may be possible when the given angle is acute and the opposite side is shorter than the adjacent side.

Cosine Rule

Finding a side: a² = b² + c² − 2bc cos A. Finding angle: cos A = (b² + c² − a²) / (2bc). Used when: three sides known (SSS). Two sides and included angle known (SAS). Example (SAS): b=5, c=8, angle A=60°. a² = 25 + 64 − 2×5×8×cos60° = 89 − 40 = 49. a = 7. Triangle area: Area = ½ab sin C (any two sides and included angle).

Graphs and Identities

Key identities: sin²θ + cos²θ = 1 (Pythagorean identity). tan θ = sin θ / cos θ. Graphs: sin and cos repeat with period 360°. Range: −1 to +1. tan repeats with period 180°, with asymptotes at ±90°. Amplitude of y = a sin(bx + c) + d: amplitude = |a|. Period = 360°/|b|. Phase shift = −c/b. Transformations of trig graphs are commonly examined at GCSE and A-level — understanding amplitude, period, and phase shift from the equation is essential.