Trigonometry: Sine Rule, Cosine Rule and Bearings

At GCSE you extend trigonometry beyond right-angled triangles to cover the sine rule, cosine rule, the formula for the area of a triangle, exact trigonometric values, and applications in 3D and bearings.

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Exact Trigonometric Values

Memorise these exact values (from equilateral and right isosceles triangles):

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3 = √3/3
45°	1/√2 = √2/2	1/√2 = √2/2	1
60°	√3/2	1/2	√3
90°	1	0	undefined

Example: Find the exact value of sin 30° × tan 60°. = (1/2) × √3 = √3/2

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SOHCAHTOA in Right-Angled Triangles (Review)

sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent

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The Sine Rule

For any triangle with sides a, b, c opposite to angles A, B, C:

a/sin A = b/sin B = c/sin C

(Or inverted: sin A/a = sin B/b = sin C/c — use this form when finding an angle.)