What you'll learn
The sine rule, cosine rule and the area formula ½ab sin C let you work with any triangle, not just right-angled ones. In this guide you will learn when to use each rule, how to find missing sides and angles, how to calculate the area of a triangle using two sides and the included angle, and how to choose the correct rule for a given problem. These are key higher-tier trigonometry skills.
Key terms and definitions
Sine rule — relates sides and their opposite angles: a/sin A = b/sin B = c/sin C.
Cosine rule — relates all three sides and one angle: a² = b² + c² − 2bc cos A.
Included angle — the angle between two known sides.
Opposite side — the side across from a given angle.
Area formula — Area = ½ab sin C, using two sides and the angle between them.
Core concepts
Labelling a triangle
Label the angles with capital letters (A, B, C) and the side opposite each angle with the matching lower-case letter (a, b, c). This labelling is essential for applying the rules correctly: side a is opposite angle A, and so on.
The sine rule
Use the sine rule when you have a side and its opposite angle, plus one more piece of information:
a/sin A = b/sin B = c/sin C
Use it to find a missing side (when you know an angle and its opposite side, plus another angle) or a missing angle (when you know two sides and one opposite angle). Turn the formula upside down (sin A / a = …) when finding an angle.
The cosine rule
Use the cosine rule when the sine rule doesn't fit — typically when you know two sides and the included angle (to find the third side), or all three sides (to find an angle):
a² = b² + c² − 2bc cos A
To find an angle, rearrange to cos A = (b² + c² − a²) / 2bc.
Choosing the right rule
- Two sides and the included angle, or three sides → use the cosine rule.
- A side with its opposite angle plus one more angle or side → use the sine rule.
Identify what you know and what you need before choosing.
Area of a triangle
The area of any triangle is ½ab sin C, where a and b are two sides and C is the included angle between them. This works for any triangle, not just right-angled ones, and only needs two sides and the angle between them.
Worked examples
Example 1: Sine rule for a side
In a triangle, A = 40°, B = 60°, side a = 8. Find side b.
a/sin A = b/sin B → b = a sin B / sin A = 8 × sin 60° / sin 40° ≈ 8 × 0.866 / 0.643 ≈ 10.8.
Example 2: Cosine rule for a side
Two sides are 5 and 7 with an included angle of 60°. Find the third side.
a² = 5² + 7² − 2(5)(7)cos 60° = 25 + 49 − 70(0.5) = 74 − 35 = 39, so a = √39 ≈ 6.24.
Example 3: Area
Find the area of a triangle with sides 6 and 9 and included angle 30°.
Area = ½ × 6 × 9 × sin 30° = ½ × 54 × 0.5 = 13.5 square units.
Common mistakes and how to avoid them
Using the wrong rule. Two sides + included angle or three sides → cosine rule; opposite side–angle pair → sine rule.
Mislabelling. The side must be opposite its matching angle.
Using the area formula without the included angle. C must be the angle between sides a and b.
Calculator in the wrong mode. Make sure it is set to degrees.
Rounding too early. Keep full accuracy until the final answer.
Exam technique for Sine and Cosine Rule
Label the triangle with matching letters first.
Identify what you know to choose the sine or cosine rule.
Rearrange correctly — flip the sine rule for angles; rearrange the cosine rule for angles.
Use ½ab sin C for area with the included angle.
Work in degrees and round only at the end.
Quick revision summary
For any triangle, label angles A, B, C with opposite sides a, b, c. Use the sine rule (a/sin A = b/sin B = c/sin C) when you have a side and its opposite angle plus one more fact — flip it to sin A/a = … to find an angle. Use the cosine rule (a² = b² + c² − 2bc cos A) when you know two sides and the included angle (to find the third side) or all three sides (to find an angle, via cos A = (b² + c² − a²)/2bc). The area of any triangle is ½ab sin C, using two sides and the included angle between them. Choose the rule by checking what you know and need; label carefully so each side matches its opposite angle; keep your calculator in degrees; and round only at the end. These three tools handle missing sides, missing angles and areas in non-right-angled triangles.