Cosine Rule: Top GCSE Maths Topics to Master
Page Contents
Cosine Rule — How to find missing sides and missing angles in non-right-angled triangles
This blog explains how to use the cosine rule to find missing sides or missing angles in non-right-angled triangles. It shows when the cosine rule is the right tool, gives the exact formulas, and walks through detailed worked examples for both a missing side and a missing angle.
There are also cosine rule worksheets based on GCSE exam questions — perfect for practising the exact steps shown below.
What is the Cosine Rule?
The cosine rule (sometimes called the law of cosines) links the lengths of the three sides of any triangle with the cosine of one of its angles. It generalises Pythagoras’ theorem so you can work with any triangle, not just right-angled ones.
Take a look at the triangle ABC below.
There are two useful forms of the cosine rule:
1. To find a side (side \( a \) opposite angle \( A \)):
$$ a^2 = b^2 + c^2 – 2bc \cdot \cos(A) $$
You can cycle the letters: e.g.$$ b^2 = a^2 + c^2 – 2ac \cdot \cos(B) $$
2. To find an angle (angle \( A \) when you know all three sides \( a, b, c \)):
$$ \cos(A) = \frac{b^2 + c^2 – a^2}{2bc} $$
When should you use the Cosine Rule?
Use the cosine rule in either of these two situations:
1. Side–Angle–Side (SAS): you know two sides and the included angle, and you need the missing side.
Example: you know \( b, c \) and angle \( A \), want \( a \).
Use $$ a^2 = b^2 + c^2 – 2bc \cdot \cos(A) $$
2. Side–Side–Side (SSS): you know all three sides and you need a missing angle.
Example: you know \( a, b, c \), want angle \( A \).
Use $$ \cos(A) = \frac{b^2 + c^2 – a^2}{2bc} $$
*Tips: Don’t use the cosine rule when you have a right angle and can use simple Pythagoras/trigonometry — cosine rule is for non-right triangles or when the information matches SAS or SSS.
Worked example A — Finding a missing side (SAS case)
Problem: In triangle ABC, \( b = 7.5 \, \text{cm}, c = 10 \, \text{cm} \) and \( \angle A = 42^\circ. \) Find side \( a \) (to 3 significant figures).
Step 1: Identify the form.
We know two sides \( b \) and \( c \) and the included angle \( A \). This is an SAS case — use the side version of the cosine rule:
$$ a^2 = b^2 + c^2 – 2bc \cdot \cos(A) $$
Step 2: Substitute numbers carefully.
$$ a^2 = 7.5^2 + 10^2 – 2(7.5)(10) \cdot \cos(42^\circ) $$
Step 3: Evaluate the cosine term: \( \cos(42^\circ) \approx 0.7431448 \)
Now compute:
$$ a^2 = 56.25 + 100 – 150 \cdot 0.7431448 \approx 44.78 $$
Step 4: Take the square root and round.
$$ a \approx \sqrt{44.78} \approx 6.69 \, \text{cm} $$
Final Answer: The missing side \( a \approx 6.69 \, \text{cm}. \)
Worked example B — Finding a missing angle (SSS case)
Problem: Triangle ABCABCABC has side lengths \( a = 9 \, \text{cm}, b = 7 \, \text{cm}, c = 5 \, \text{cm}. \) Find angle \( A \) to the nearest degree.
Step 1: Identify the form.
We know all three sides (SSS). Use the angle form of the cosine rule:
$$ \cos(A) = \frac{b^2 + c^2 – a^2}{2bc} $$
Step 2: Substitute numbers:
$$ \cos(A) = \frac{7^2 + 5^2 – 9^2}{2(7)(5)} $$
$$ \cos(A) = \frac{-7}{70} \approx -0.1 $$
Step 3: Find the angle using inverse cosine.
$$ A = \cos^{-1}(-0.1) $$
To the nearest degree, \( A \approx 96^\circ. \)
Final Answer: The missing angle \( A \approx 96^\circ. \)
Additional practical tips
- Calculator mode — ensure your calculator is in degrees when working with GCSE/most school problems unless the question explicitly uses radians.
- Sign of cosine — if \( \cos \) value is negative, the corresponding angle is obtuse (>90°); if positive, the angle is acute (<90°). This helps sanity-check results.
- Round at the end — keep extra precision in intermediate steps; only round the final answer to the requested format (e.g. 2 d.p., 3 s.f., nearest degree).
- Which formula first? — always decide whether you need a missing side (use \( a^2 = b^2 + c^2 – 2bc \cdot \cos(A) \)) or a missing angle (use \( \cos(A) = \frac{b^2 + c^2 – a^2}{2bc} \)). Choosing correctly saves time.
Practise and resources
To gain fluency, practise both SAS and SSS problems across a range of numbers and shapes. There are also cosine rule worksheets based on GCSE exam questions that combine cosine rule with other topics (e.g. area, sine rule, bearings) — these are excellent for exam-style practice and improving problem selection skills.
Summary
- The cosine rule works for any triangle and is the go-to tool for SAS (find a missing side) and SSS (find a missing angle) cases.
- Use \( a^2 = b^2 + c^2 – 2bc \cdot \cos(A) \) for missing sides, and \( \cos(A) = \frac{b^2 + c^2 – a^2}{2bc} \) for missing angles.
- Follow the worked examples above step-by-step and practise with varied questions — this builds confidence and exam readiness.
See how Think Academy helps achieve top grades in GCSE exams, and book a trial for free:
