Naming sides and angles
Both rules use a shared labelling habit: each side is named with the lower-case letter that matches the upper-case angle directly opposite it. So side \(a\) is opposite angle \(A\), side \(b\) is opposite angle \(B\), and so on.
- Opposite pair
- An angle and the side facing it across the triangle, such as \(A\) and \(a\). The rules connect these pairs, so getting the labels right is half the work.
With that in place, the two rules cover every non right-angled case you meet at this level.
The sine rule
The sine rule connects each side with the sine of its opposite angle. Every fraction in it is equal:
Use it when your known and unknown pieces form opposite pairs: you need one complete pair (a side and its opposite angle) plus one more piece from a second pair. Take just the two fractions you need and cross-multiply.
In a triangle, angle \(A = 40^\circ\), angle \(B = 75^\circ\), and side \(a = 8\) cm. Find side \(b\), to 1 decimal place.
The cosine rule
The cosine rule handles the cases the sine rule cannot: it uses all three sides and one angle, so no opposite pair is needed. To find a side opposite a known angle \(A\):
The same formula rearranges to find an angle when you know all three sides. Use the cosine rule when you have two sides and the angle between them, or when you have all three sides.
A triangle has sides \(b = 7\) cm and \(c = 9\) cm with the angle \(A = 50^\circ\) between them. Find side \(a\), to 1 decimal place.
Follow the order of operations
In the cosine rule, work out \(2bc\cos A\) as one number first, then subtract it. Do not square-root until the whole right-hand side is a single value, or you will lose the answer.
Choosing which rule
Look at what the question gives you, then match it to this table. The letters describe the pattern: S for a known side, A for a known angle.
| What you are given | Pattern | Rule to use |
|---|---|---|
| A side with its opposite angle, plus one more of either | Opposite pair | Sine rule |
| Two sides and the angle between them | SAS | Cosine rule |
| All three sides (find an angle) | SSS | Cosine rule |
Where this is assessed
Selecting the right rule and carrying out the calculation is Criterion A (Knowing and understanding). Laying out the chosen rule, the substitution and the rounding is Criterion C (Communicating). Using it for bearings, surveying and distances that cannot be measured directly is Criterion D (Applying mathematics in real-life contexts). A quick test: if you can spot a complete opposite pair, start with the sine rule; if not, it is the cosine rule.
Check yourself
1. A triangle has sides 5 cm, 6 cm and 7 cm. Find the largest angle, to 1 decimal place. +
Three sides means the cosine rule, and the largest angle faces the longest side (7 cm). Rearranged, \(\cos C = \dfrac{5^2 + 6^2 - 7^2}{2(5)(6)} = \dfrac{25 + 36 - 49}{60} = \dfrac{12}{60} = 0.2\). So \(C = \cos^{-1}(0.2) \approx \mathbf{78.5^\circ}\).
2. Angle \(A = 35^\circ\), angle \(B = 60^\circ\), and side \(a = 9\) cm. Find side \(b\), to 1 decimal place. +
The complete pair \(A, a\) with angle \(B\) means the sine rule: \(\dfrac{9}{\sin 35^\circ} = \dfrac{b}{\sin 60^\circ}\). So \(b = \dfrac{9 \times \sin 60^\circ}{\sin 35^\circ} = \dfrac{9 \times 0.8660\ldots}{0.5735\ldots} \approx \mathbf{13.6}\) cm.
3. You know two sides of a triangle and the angle between them. Which rule finds the third side? +
Two sides and the included angle (SAS) is a cosine rule case: use \(a^2 = b^2 + c^2 - 2bc\cos A\). The sine rule cannot start here because there is no complete opposite pair to build on.
Part of the Standard Maths library. Spotted an error or want a topic added? That feedback makes the notes better.