Sine rule, cosine rule and area of a non-right-angled triangle
These rules extend trigonometry beyond right-angled triangles. They appear on CCEA Paper 2 (calculator) and are assessed at Higher tier (and sometimes Foundation tier for the basic area formula).
When to use which rule
| Given information | Use |
|---|---|
| Two sides + one opposite angle (or two angles + one side) | Sine rule |
| Two sides + the included angle (SAS) | Cosine rule (find third side) |
| All three sides (SSS) | Cosine rule (find an angle) |
| Two sides + any angle + area needed | Area formula |
The sine rule
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Or equivalently (for finding angles):
$$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$$
Where a, b, c are sides opposite angles A, B, C respectively.
Finding a side: a = b × sin A / sin B. Finding an angle: sin A = a × sin B / b, then A = sin⁻¹(...).
The ambiguous case (Higher): when given two sides and a non-included angle (SSA), there may be two possible triangles. Check if the second angle is also valid.
The cosine rule
Finding a side: a² = b² + c² − 2bc cos A.
Finding an angle: cos A = (b² + c² − a²) / (2bc).
Note: the cosine rule reduces to Pythagoras when A = 90° (cos 90° = 0).
Area of a triangle
$$\text{Area} = \frac{1}{2}ab \sin C$$
Where a and b are two known sides and C is the included angle (the angle between those two sides).
Example: two sides of 8 cm and 11 cm with included angle 42°. Area = ½ × 8 × 11 × sin 42° = 44 × 0.6691... ≈ 29.4 cm².
CCEA examiner context
CCEA Paper 2 often presents these in context: bearings, navigation, surveying, triangular fields. Always draw a diagram and label the sides and angles clearly before choosing a rule.
⚠Common mistakes
- Using SOHCAHTOA on non-right-angled triangles — it only works on right-angled triangles.
- Misidentifying the included angle in the cosine rule or area formula.
- Rounding intermediate answers: keep full calculator accuracy until the final answer.
- Ambiguous case: forgetting there may be two triangles (the obtuse solution).
- Cosine rule for angle: a common error is writing (a² − b² − c²) / 2bc instead of (b² + c² − a²) / 2bc.
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