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.)

Finding a side: Example: B = 73°, C = 41°, b = 14 cm. Find c. A = 180° − 73° − 41° = 66° c/sin 41° = 14/sin 73° → c = 14 sin 41°/sin 73° ≈ 9.65 cm

Finding an angle: Example: a = 8 cm, b = 11 cm, A = 35°. Find B. sin B/11 = sin 35°/8 → sin B = 11 sin 35°/8 ≈ 0.789 → B ≈ 52.0° or 128.0° (ambiguous case)

The Ambiguous Case

When finding an angle using the sine rule, there may be two possible answers (one acute, one obtuse). Check whether the obtuse angle is valid given the other angles in the triangle.

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The Cosine Rule

Finding a side: a² = b² + c² − 2bc cos A

Example: b = 7 cm, c = 10 cm, A = 55°. Find a. a² = 49 + 100 − 2(7)(10)cos 55° = 149 − 140 × 0.5736 = 149 − 80.3 = 68.7 a = √68.7 ≈ 8.29 cm

Finding an angle: cos A = (b² + c² − a²)/(2bc)

Example: a = 9 cm, b = 7 cm, c = 5 cm. Find angle A. cos A = (49 + 25 − 81)/70 = −7/70 = −0.1 A = cos⁻¹(−0.1) ≈ 95.7°

A negative cosine value indicates an obtuse angle (between 90° and 180°).

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Area of a Triangle Using Trigonometry

Area = (1/2) ab sin C

where a and b are two sides and C is the included angle between them.

Example: Two sides 8 cm and 11 cm, included angle 65°. Area = (1/2)(8)(11) sin 65° = 44 × 0.906 ≈ 39.87 cm²

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Choosing the Right Rule