Watch out
Using sine rule without a matching pair.
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SQA National 5 Mathematics
Cosine rule
Solving non-right-angled triangles when the included angle or all sides are known.
Formula reference
Check the National 5 rules and formulae linked to this topic.
Which method?
Match exam clues to a suitable method.
Before you start
- Check the triangle is not right-angled.
- Use cosine rule for SAS or SSS information.
- Label the side opposite the angle you are using.
Explanation
The cosine rule is used in non-right-angled triangles when sine rule is not suitable.
Use a² = b² + c² − 2bc cos A to find a side when two sides and the included angle are known.
Use the rearranged cosine rule to find an angle when all three sides are known.
Visual support
Included angle A
Key formulae and rules
- a² = b² + c² − 2bc cos A
- cos A = (b² + c² − a²) / 2bc
Check
Check your answer against the diagram. The longest side should still be opposite the largest angle.
Exam tip
Cosine rule is the default when you have SAS or SSS and no right angle.
Calculator tip
Make sure the calculator is in degree mode before using sin, cos, tan or inverse trig.
Worked examples
Worked example 1
Find a side
Find side a when b = 9 cm, c = 12 cm and A = 60 degrees.
- a² = 9² + 12² − 2(9)(12)cos 60.
- a² = 81 + 144 − 108 = 117.
- a = √117.
Answer: a = 10.8 cm to 1 decimal place.
Worked example 2
Find an angle
A triangle has sides a = 7 cm, b = 9 cm, c = 11 cm. Find angle A.
- cos A = (9² + 11² − 7²) / (2(9)(11))
- cos A = 153 / 198
- A = cos−1(153 / 198).
So: A = 39.3 degrees to 1 decimal place.
Worked example 3
Choose cosine rule
Two sides are 5 cm and 8 cm with included angle 110 degrees. Find the third side.
- The included angle is given, so use cosine rule.
- x² = 5² + 8² − 2(5)(8)cos 110
- x² = 116.36..
Answer: x = 10.8 cm to 1 decimal place
Watch out
- Using sine rule without a matching pair.
- Using the wrong angle in the cosine rule.
- Forgetting the negative sign before 2bc cos A.