For students and parents · O-Level E-Math
Sine Rule or Cosine Rule: Which One, and When
Choosing the right rule before solving a triangle.
Mrs Eileen Toh
Founder & Curriculum Architect · ex-MOE · 3 min read · Updated 29 Jun 2026
In the solution of triangles, the mistake is almost never the arithmetic. It is reaching for the wrong rule. A student who picks the cosine rule when the sine rule was faster, or the sine rule when there was no matched pair to use, has lost the question before doing any sums. So the skill worth training is not the formulas, which are on the sheet, but the choice between them.
GPA teaches that choice as a short routine you run before touching the calculator.
The three tools
The sine rule: a/sin A = b/sin B = c/sin C. Use it when you have a matched pair, meaning a side opposite a known angle, plus one more side or angle.
The cosine rule: c2 = a2 + b2 - 2ab cos C. Use it when you have two sides and the included angle, to find the third side, or all three sides, to find an angle.
The area of a triangle: (1/2) ab sin C. Use it when you have two sides and the angle between them.
The decision, in one question
Before you start, ask one thing: do I have a matched pair, a side opposite a known angle?
If yes, use the sine rule.
If no, you have either two sides and the included angle, or all three sides, and both of those are cosine rule. Two sides and the included angle also gives you the area straight away.
That single question sorts almost every triangle in the syllabus.
Worked example A, the sine rule
A triangle has A = 40 degrees, B = 75 degrees, and side a = 8 opposite A. Find side b.
There is a matched pair, side a opposite angle A, so the sine rule fits:
b = a x sin B / sin A = 8 x sin 75 / sin 40 = 12.0 to 3 significant figures.
Worked example B, the cosine rule and the area
A triangle has sides a = 7 and b = 10 with the included angle C = 60 degrees. Find side c and the area.
There is no side opposite a known angle yet, but there are two sides and the angle between them, so it is the cosine rule:
c2 = 72 + 102 - 2(7)(10) cos 60 = 49 + 100 - 140(0.5) = 79, so c = √(79) = 8.89 to 3 significant figures.
And because you have two sides and the included angle, the area comes free:
Area = (1/2)(7)(10) sin 60 = 35 x √(3)/2 = 30.3 to 3 significant figures.
When all three sides are known
If you are given all three sides and asked for an angle, that is still the cosine rule, rearranged. For sides 5, 6 and 7, the angle opposite the longest side is cos-1((52 + 62 - 72)/(2 x 5 x 6)) = 78.5 degrees. No matched pair, so no sine rule; three sides, so cosine rule.
The teaching point
The three-question routine, is there a right angle, is there a matched pair, or is it two sides and the included angle, decides the tool before any button is pressed. Once the tool is chosen the sums are short. This is the same diagnose-first discipline GPA carries through the whole of Secondary math: classify the question, then compute.
If your child can use both rules but keeps choosing the wrong one, that is exactly the gap GPA's Secondary Math programme is built to close, working from real E-Math papers. A short diagnostic consult will show where the choosing breaks down.
Every worked value on this page was checked independently before publishing.