The sine rule & the cosine rule
For non-right triangles the sine rule a/sin A = b/sin B = c/sin C pairs a side with its opposite angle, while the cosine rule a² = b² + c² − 2bc cos A handles the cases the sine rule cannot. The choice is mechanical: use the cosine rule when you know three sides (SSS) or two sides and the included angle (SAS), and the sine rule otherwise.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Label consistentlySide a is opposite angle A, and so on. Mislabelling breaks both rules.
- Choose by what you are givenSSS or SAS means cosine rule. ASA, AAS or SSA means sine rule.
- Rearrange the cosine rule for an anglecos A = (b² + c² − a²)/(2bc). The largest side is opposite the largest angle, which may be obtuse.
- Check the ambiguous SSA caseThe sine rule can give two valid triangles, since sin θ = sin(180° − θ). Test whether the obtuse option keeps the angle sum below 180°.
Worked example
A triangle has sides b = 5, c = 7 and included angle A = 60°. Find a.
- Two sides and the included angle: SAS, so use the cosine rule.
- a² = b² + c² − 2bc cos A = 25 + 49 − 2(5)(7)cos 60°.
- cos 60° = 1/2, so the last term is 70 × 1/2 = 35.
- a² = 74 − 35 = 39.
Answer. a = √39 ≈ 6.24 — between 5 and 7, as expected for a 60° included angle.
Where marks get dropped
These are the specific errors that cost credit on the sine rule & the cosine rule questions — QED's rubric penalises each of them separately.
- Using the sine rule for SAS. Neither known angle is opposite a known side, so the sine rule cannot start.
- Ignoring the ambiguous case in SSA. Always check whether 180° − your angle also gives a valid triangle.
- Getting the cosine rule sign wrong. It is MINUS 2bc cos A, which reduces to Pythagoras when A = 90°.
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The sine rule & the cosine rule — frequently asked questions
When is the ambiguous case a real risk?
In SSA, when the side opposite the known angle is shorter than the other known side but longer than the perpendicular height. Then two triangles exist.
How does the cosine rule relate to Pythagoras?
With A = 90°, cos A = 0 and it collapses to a² = b² + c². It is Pythagoras with a correction term for non-right angles.
Which rule finds the area?
Neither directly — use Area = ½ab sin C with two sides and the included angle, or Heron’s formula from three sides.
The rest of Trigonometry
Triangles, the unit circle, identities and periodic functions. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Right-triangle ratios: sin, cos & tan
- 2The unit circle & exact values
- 3Radians, degrees & arc length
- 4Graphs of sin, cos & tan
- 5Pythagorean & reciprocal identities
- 6Angle-sum, difference & double-angle formulas
- 7Solving trigonometric equations
- 8The sine rule & the cosine rule
- 9Triangle area, sectors & segments
- 10Inverse trigonometric functions
- 11Proving trigonometric identities
- 12Polar coordinates & converting to Cartesian
- 13Modelling periodic phenomena with sinusoids
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