What you'll learn
The sine rule and cosine rule extend trigonometry beyond right-angled triangles to any triangle, and they are an important part of the CSEC Mathematics syllabus under Geometry and Trigonometry. Right-angled trigonometry (SOH CAH TOA) only works when the triangle has a 90° angle; the sine and cosine rules let you find unknown sides and angles in scalene and obtuse triangles, which appear constantly in bearings, surveying, navigation and structural problems. In this guide you will learn when to use each rule, how to apply them accurately, how to find the area of a triangle using ½ab sin C, and how to combine these tools in multi-step Paper 2 problems. Questions involving these rules are common in Paper 2 and frequently carry several marks.
Key terms and definitions
Sine rule — the relationship a ÷ sin A = b ÷ sin B = c ÷ sin C, linking each side to the sine of its opposite angle.
Cosine rule — the relationship a² = b² + c² − 2bc cos A (and its rearrangements), used for the third side or for an angle.
Included angle — the angle between two known sides.
Opposite side — the side facing a given angle; side a is opposite angle A.
Scalene triangle — a triangle with all three sides (and angles) different.
Bearing — a direction measured clockwise from north, written as three figures (e.g. 075°).
Area of a triangle — given two sides and the included angle, Area = ½ ab sin C.
Core concepts
Labelling a triangle
By convention, the vertices (corners) are labelled with capital letters A, B and C, and the side opposite each vertex takes the matching lower-case letter: side a is opposite angle A, side b opposite angle B, and side c opposite angle C. Getting this labelling right is the foundation for both rules, so always mark your diagram before substituting.
The sine rule
Use the sine rule when you have a side and its opposite angle as a complete pair, plus one more piece of information. The rule is:
a ÷ sin A = b ÷ sin B = c ÷ sin C.
To find a side, keep the sides on top: a ÷ sin A = b ÷ sin B. To find an angle, it is easier to flip the rule: sin A ÷ a = sin B ÷ b. The sine rule is the correct tool in two situations: (1) you know two angles and any one side (because the third angle is easily found, then any side follows), or (2) you know two sides and an angle opposite one of them.
The cosine rule
Use the cosine rule when the sine rule cannot start — that is, when you do not have a complete side–opposite-angle pair. There are two cases. To find a side when you know two sides and the included angle: a² = b² + c² − 2bc cos A. To find an angle when you know all three sides, rearrange: cos A = (b² + c² − a²) ÷ (2bc). The cosine rule reduces to Pythagoras' theorem when the angle is 90°, because cos 90° = 0.
Choosing between the rules
A quick decision process: if you can pair a known side with its opposite known angle, start with the sine rule. If you only know three sides (SSS) or two sides and the angle between them (SAS), you must use the cosine rule. Often a Paper 2 problem needs the cosine rule first to find one quantity, then the sine rule to finish.
Area of a triangle
When you know two sides and the included angle, the area is Area = ½ ab sin C, where C is the angle between sides a and b. This avoids needing the perpendicular height. It is widely used in land-area and field problems in the Caribbean context.
Bearings and applications
Bearings problems usually translate a real direction into a triangle. Draw the north lines, mark the angles carefully, and identify the triangle you must solve. Then apply the sine or cosine rule as appropriate. Neat, large diagrams prevent most errors here.
Worked examples
Example 1: Sine rule for a side (Paper 2 style)
In triangle ABC, angle A = 40°, angle B = 65°, and side a = 12 cm. Find side b.
By the sine rule, b ÷ sin B = a ÷ sin A, so b = a × sin B ÷ sin A = 12 × sin 65° ÷ sin 40°. Since sin 65° ≈ 0.9063 and sin 40° ≈ 0.6428, b ≈ 12 × 0.9063 ÷ 0.6428 ≈ 16.9 cm (3 s.f.).
Example 2: Cosine rule for a side (Paper 2 style)
Two roads leave a junction at an angle of 70°. A surveyor walks 50 m along one road and 80 m along the other. How far apart are the endpoints?
This is SAS: two sides (50 and 80) with the included angle (70°). Let the unknown distance be a, with b = 50, c = 80, A = 70°. Then a² = 50² + 80² − 2(50)(80)cos 70° = 2500 + 6400 − 8000 × 0.3420 ≈ 8900 − 2736 = 6164. So a ≈ √6164 ≈ 78.5 m (3 s.f.).
Example 3: Cosine rule for an angle (Paper 2 style)
A triangular plot has sides 7 m, 9 m and 12 m. Find the largest angle.
The largest angle faces the longest side (12 m), so let a = 12, b = 7, c = 9. Then cos A = (b² + c² − a²) ÷ (2bc) = (49 + 81 − 144) ÷ (2 × 7 × 9) = (−14) ÷ 126 ≈ −0.1111. So A = cos⁻¹(−0.1111) ≈ 96.4°. The negative cosine correctly gives an obtuse angle.
Common mistakes and how to avoid them
Using right-angle trig on a non-right triangle. SOH CAH TOA only works with a 90° angle. If there is no right angle, reach for the sine or cosine rule.
Picking the wrong rule. No complete side–opposite-angle pair? You cannot start with the sine rule — use the cosine rule.
Calculator in the wrong mode. CSEC works in degrees. Check your calculator shows "DEG" (or "D"), not radians, before every trig question.
Rounding too early. Keep full accuracy in intermediate steps and round only the final answer to the required significant figures.
Forgetting the included angle for area. Area = ½ ab sin C needs the angle between the two sides, not any angle.
Mislabelling sides and angles. Always draw and label the triangle so that side a sits opposite angle A.
Exam technique for the Sine and Cosine Rules
Draw and label first. A clear diagram with correct a/A, b/B, c/C labelling prevents most mistakes and is often worth a mark itself.
State the rule before substituting. Writing "by the cosine rule, a² = b² + c² − 2bc cos A" earns a method mark even if the arithmetic later slips.
Decide the rule from what you are given. SSS or SAS → cosine rule; a matching side–angle pair → sine rule.
Find the largest angle opposite the longest side. This helps you check whether an obtuse angle is expected (cosine will be negative).
Round only at the end, to the accuracy the question demands (usually 3 significant figures or 1 decimal place).
Quick revision summary
The sine and cosine rules solve triangles that are not right-angled. Label each side opposite its matching angle (a opposite A, and so on). The sine rule, a ÷ sin A = b ÷ sin B = c ÷ sin C, is used when you have a complete side-and-opposite-angle pair plus one more fact; flip it to sin A ÷ a = … when finding an angle. The cosine rule, a² = b² + c² − 2bc cos A, is used for SAS (two sides and the included angle, to find the third side) or SSS (three sides, rearranged to cos A = (b² + c² − a²) ÷ 2bc, to find an angle). The area of any triangle is ½ ab sin C using two sides and the included angle. Keep your calculator in degree mode, draw a labelled diagram, state the rule before substituting, keep full accuracy until the final rounding, and check that an obtuse angle produces a negative cosine. These tools are essential for bearings, navigation and land-measurement problems.