Cosine Rule
Practise cosine rule with this free Higher GCSE Maths worksheet from Virtus Academy. You'll work through applying the sine rule, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. Use the cosine rule with two sides and the included angle, or all three sides.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.
Topic overview
The cosine rule handles triangles where the sine rule cannot help. It states \(a^2 = b^2 + c^2 - 2bc\cos A\), where \(A\) is the angle between sides \(b\) and \(c\).
Use it in two situations: when you know two sides and the angle between them and want the third side, or when you know all three sides and want an angle.
It is a generalisation of Pythagoras. If \(A = 90^\circ\) then \(\cos A = 0\), the last term vanishes, and the formula becomes \(a^2 = b^2 + c^2\) exactly. Recognising this makes the rule easier to trust and remember.
Revision notes
Finding a side
Substitute the two known sides and the included angle, then take the square root at the end.
With \(b = 7\), \(c = 9\) and \(A = 50^\circ\): \(a^2 = 49 + 81 - 2(7)(9)\cos 50 = 49.0\), so \(a = 7.0\)cm.
Finding an angle
Rearrange to \(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\), then use the inverse cosine.
With sides \(5\), \(7\) and \(9\), finding the angle opposite \(9\): \(\cos A = \frac{25 + 49 - 81}{70}\), giving \(A = 95.7^\circ\).
The link with Pythagoras
When the angle is \(90^\circ\), \(\cos 90 = 0\), so the final term disappears.
The rule then reduces to \(a^2 = b^2 + c^2\), which is Pythagoras' theorem.
Key points
- \(a^2 = b^2 + c^2 - 2bc\cos A\).
- \(A\) is the angle between sides \(b\) and \(c\).
- Use it for two sides and the included angle.
- Use it when all three sides are known.
- Rearrange to \(\cos A = \frac{b^2+c^2-a^2}{2bc}\) for an angle.
- It reduces to Pythagoras when the angle is \(90^\circ\).
Worked examples
Example 1
Find side \(a\) when \(b = 6\), \(c = 8\) and \(A = 60^\circ\).
Working
Example 2
Find the angle opposite the side of length \(7\) in a triangle with sides \(5\), \(6\) and \(7\).
Working
Example 3
Explain why the cosine rule becomes Pythagoras when \(A = 90^\circ\).
Working
Common mistakes
Using the wrong angle.
A must be the angle between the two sides b and c, not any other angle.
Forgetting the square root.
The formula gives a², so the final step is taking the root.
Subtracting in the wrong order when finding an angle.
The numerator is b² + c² − a², where a is opposite the angle wanted.
Using the sine rule when there is no matching pair.
Two sides and the included angle always needs the cosine rule.
Exam tips
- Identify the included angle before substituting.
- Work out each term separately to avoid calculator errors.
- Remember the final square root when finding a side.
- Use the inverse cosine when finding an angle.
Key terms
- Cosine rule
- A rule relating all three sides and one angle.
- Included angle
- The angle between two known sides.
- Inverse cosine
- The function giving an angle from a cosine value.
- Generalisation
- A rule that includes a simpler one as a special case.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.