Sheet № 50 · Higher only · AQA · Edexcel · OCR
Sine & Cosine Rules –
The sine and cosine rules extend trigonometry beyond right-angled triangles — and they are among the most important topics on the GCSE Higher paper. While SOHCAHTOA only works when there is a 90° angle, the sine and cosine rules work for any triangle. AQA, Edexcel and OCR all include these rules on their Higher papers, often as 4- or 5-ma
§Key definitions
Step 1:
Use the sine rule: p/sin P = q/sin Q.
Step 2:
9/sin 42° = q/sin 73°.
Step 3:
9/0.6691 = q/0.9563.
Step 4:
13.451 = q/0.9563.
Step 5:
q = 13.451 × 0.9563 = 12.9 cm (1 d.p.).
§Formulas to memorise
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}
a^2 = b^2 + c^2 - 2bc\cos A
\cos A = \frac{b^2 + c^2 - a^2}{2bc}
\text{Area} = \frac{1}{2}ab\sin C
\text{Area} = \frac{1}{2} \times 11 \times 8 \times \sin 55° = 44 \times 0.8192 = \textbf{36.0 cm}^2 \text{ (1 d.p.)}
\cos A = \frac{7^2 + 9^2 - 12^2}{2 \times 7 \times 9} = \frac{49 + 81 - 144}{126} = \frac{-14}{126} = -0.1111
A = \cos^{-1}(-0.1111) = \textbf{96.4°} \text{ (1 d.p.)}
Worked example
See example below.
In triangle PQR, angle P = 42°, angle Q = 73° and side p (opposite P) = 9 cm. Find side q (opposite Q).
⚠ Common mistakes
- ✗Using the wrong rule. If you have the included angle and two sides, use cosine. If you have a complete angle-side pair, use sine. Choosing the wrong one makes the question impossible.
- ✗Mislabelling sides and angles. Side a must be opposite angle A, side b opposite angle B, and so on. Getting this wrong produces an incorrect answer.
- ✗Forgetting to square root in the cosine rule. The formula gives a² — do not forget the final square root.
- ✗Not recognising the obtuse angle. If cos A is negative, the angle is obtuse. Some students panic — this is perfectly normal and expected.
- ✗The ambiguous case. When using the sine rule to find an angle (SSA), there can be two possible triangles. At GCSE this is rare but be aware of it.
✦ Exam tips
- →Decide which rule to use before writing anything. Draw a quick sketch, label the sides and angles, and check: do I have SAS or SSS (cosine rule) or a matching pair (sine rule)?
- →Copy the formula from the formula sheet into your working. This shows the examiner which rule you are applying.
- →For area questions, use ½ab sin C rather than base × height when you do not have a perpendicular height.
- →Show intermediate values — examiners award marks for the substitution and for the calculation before the final answer.
- →If finding all angles, use the cosine rule for the first angle, then the sine rule (or angle sum) for the rest.