Use the sine rule when you have a matched angle-side pair (you know one angle and the side opposite it) plus one more angle or side. Use the cosine rule when you have two sides and the included angle (SAS), or all three sides (SSS). If the question gives you a right angle, use basic trigonometry (SOH-CAH-TOA) and Pythagoras' theorem first, since that's faster. This single decision flowchart covers every non-right-angled triangle question in O-Level E Maths.
Sine rule
a/sin A = b/sin B = c/sin C
or inverted:
sin A/a = sin B/b = sin C/c
Cosine rule
a² = b² + c² − 2bc cos A
or rearranged:
cos A = (b²+c²−a²) / 2bc
The sine rule is simpler to use — one multiplication step. The cosine rule takes more working. So your default should always be: can I use the sine rule? Only switch to the cosine rule when you have to.
Every non-right-angled triangle question gives you some combination of sides and angles. Read off what you're given, then follow the chart.
This is the key concept. In triangle ABC, side a is the side opposite angle A. They are a matched pair. Side b is opposite angle B, and side c is opposite angle C.
A matched pair exists when you know both the angle and the length of the side sitting opposite it. If the question gives you angle B = 48° and side b = 9 cm, you have a matched pair. You can now use the sine rule with any other angle or side in the triangle.
The most common confusion
Students see two sides and an angle and reach for the sine rule. But if the angle is between the two sides (not opposite one of them), that is SAS — cosine rule. The angle has to be opposite one of your known sides for the sine rule to apply.
In triangle PQR, angle P = 62°, angle Q = 47°, and side p = 14 cm. Find the length of side q.
Solution
p / sin P = q / sin Q14 / sin 62° = q / sin 47°q = 14 × sin 47° / sin 62°q = 14 × 0.7314 / 0.8829 = 11.6 cm (3 s.f.)In triangle ABC, AB = 7 cm, AC = 10 cm, and angle A = 55°. Find BC.
Solution
a² = b² + c² − 2bc cos Aa² = 10² + 7² − 2(10)(7) cos 55°a² = 100 + 49 − 140 × 0.5736a² = 149 − 80.30 = 68.70a = √68.70 = 8.29 cm (3 s.f.)In triangle ABC, a = 8 cm, b = 11 cm, c = 6 cm. Find angle A.
Solution
cos A = (b² + c² − a²) / 2bccos A = (11² + 6² − 8²) / (2 × 11 × 6)cos A = (121 + 36 − 64) / 132cos A = 93 / 132 = 0.7045A = cos⁻¹(0.7045) = 45.2° (1 d.p.)After finding one angle from SSS, switch to the sine rule. Once you have one angle, you now have a matched pair (the angle and the side opposite it). The sine rule is faster for finding the remaining angles — avoid using the cosine rule three times.
If the question asks for the area of a non-right-angled triangle, always use:
Area = ½ ab sin C
Here, a and b are any two sides, and C is the angle between them (the included angle). This requires SAS — the same setup as the cosine rule. If a question gives you SAS, expect it to ask for both a missing side (cosine rule) and the area (½ab sin C) in separate parts.
When the sine rule gives you sin A = 0.73, there are two possible angles: one acute (about 47°) and one obtuse (about 133°). The calculator gives the acute one because it treats trigonometric ratios of an obtuse angle the same way as an acute one. You must decide which is correct using the context of the triangle — a triangle cannot have two obtuse angles, so if another angle is already large, the missing one must be acute.
Examiner trap
If the question says "angle A is obtuse" and the sine rule gives sin A = 0.73, the answer is not 47°. It is 180° − 47° = 133°. Missing this loses the mark even if all your working is correct.
Can I always use the cosine rule instead of the sine rule?
Technically yes — but the cosine rule produces much more complex algebra when you only have AAS or ASA. The sine rule is a one-step rearrangement. For exam efficiency, use the sine rule whenever you have a matched pair.
When would I use the sine rule to find an angle?
When you have SSA: two sides and an angle that is not between those sides. You have a matched pair, so you can write sin A / a = sin B / b and solve for the unknown angle. This is also the scenario that can produce an obtuse angle — always check.
The question has a right angle — do I still use these?
No. With a right angle, use SOH-CAH-TOA and Pythagoras. Sine and cosine rules still work mathematically on right-angled triangles, but they take more steps. Only switch to sine/cosine rule when there is definitely no right angle.
What does "included angle" mean?
The included angle is the angle formed at the vertex where the two known sides meet. In triangle ABC, if you know sides AB and AC, the included angle is angle A — it sits between those two sides. If you know sides AB and BC, the included angle is angle B.
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