National 5 Maths
The Cosine Rule
Page sections 
- Topic content
- Textbook page numbers
- Key ideas
- Worked examples
- Past paper questions
- Worksheets
- Notes and videos
Topic content
- Use the cosine rule to find either a side or an angle in a triangle.
- If finding a side, you will always be given the "included angle" (the angle between the other two sides).
Textbook page numbers
- Zeta National 5+ Maths pp.228-230
- TeeJay Maths Book N5 pp.81-84
- Leckie National 5 Maths pp.302-305
Buy our favourite textbook
Zeta National 5+ Maths
Clear and comprehensive.
Progressive exercises.
Includes answers.
Buy from Zeta Press 
Key ideas
- The cosine rule works in non-right angled triangles.
- We use the cosine rule when the question involves all three sides and one angle.
- There are two types of questions:
• Find a side, given the other two sides and their included angle.
• Find any angle, given all three sides. - Angles are given using capital letters: A°, B°, C°, etc. Side \(a\) is opposite A°, side \(b\) is opposite B°, etc.
- The cosine rule is on the formulae list in two versions.
- When we are finding side \(a\), we use: $$ \large a^2=b^2+c^2-2bc\ cos\ A $$
- When we are finding angle \(A\), we can use either the first version or: $$ \large cos\ A=\frac{b^2+c^2-a^2}{2bc} $$
- Nat 5 cosine rule and sine rule questions are often combined with bearings or related angles.
Looking for an N5 Maths tutor?
Try our free, no-obligation tutor search tool.
Click here to find a tutor in your area. 
Example 1 (non-calculator)
In the triangle below, XY = 7 cm, XZ = 9 cm and cos X = \(\large\frac{1}{9}\small.\)
Calculate the length of YZ, expressing your answer as a surd in its simplest form.
Example 2 (calculator)
In the triangle below, RT = 1.3 cm, ST = 2.9 cm and angle RTS = 108°.
Calculate the length of RS, correct to 3 significant figures.
Example 3 (calculator)
In the triangle below, AB = 12 mm, BC = 15 mm and AC = 7 mm.
Calculate the size of angle ACB.
Example 4 (calculator)
A triangle has sides of length 5.4 cm, 7 cm and 9.1 cm. Calculate its smallest angle.
Example 5 (calculator)
Ruby is practising ruler-and-compass constructions.
She uses a sharp pencil and carefully rules a line of length 4.8 cm. She then uses her compasses to measure an angle of 37° at one end of the line. Finally, she uses her ruler to draw another line, of length 6.3 cm.
If she has measured her two lengths and the angle accurately, calculate how far apart the ends of her two lines should be, correct to the nearest millimetre.
Example 6 (calculator)
SQA National 5 Maths 2015 P2 Q3
Triangle ABC is shown below.
Calculate the length of AB.
Example 7 (calculator)
SQA National 5 Maths 2017 P2 Q3
A piece of land is in the shape of a triangle as shown.
The diagram shows triangle JKL.
• PQ = 250 metres
• PR = 180 metres
• angle QPR = 147°
The owner wishes to build a fence along the side QR.
Calculate the length of the fence.
Example 8 (non-calculator)
SQA National 5 Maths 2018 P1 Q10
In triangle XYZ:
• XZ = 10 centimetres
• YZ = 8 centimetres
• cos Z = \(\large\frac{1}{8}\)
Calculate the length of XY.
Example 9 (calculator)
SQA National 5 Maths 2019 P2 Q7
Triangle XYZ is shown below.
Calculate the size of the smallest angle in triangle XYZ.
Example 10 (non-calculator)
SQA National 5 Maths 2023 P1 Q6
In triangle ABC:
• AC = 5 metres
• BC = 6 metres
• cos C = \(\large\frac{1}{5}\)
Calculate the length of AB.
Buy N5 Maths practice papers
Zeta: Five Practice Papers TOP CHOICECGP: N5 Maths Exam Practice
Leckie: Revision and Practice
Hodder: N5 Maths Practice Papers
Past paper questions
| • All past paper questions by topic |
|
Finding a side: • 2015 Paper 2 Q3 • 2016 P2 Q16 (with SOH CAH TOA) • 2017 Specimen Paper 2 Q5 • 2017 Paper 2 Q3 • 2018 Paper 1 Q10 • 2021 Paper 2 Q7 (with bearings) • 2023 Paper 1 Q6 |
| Finding an angle: • 2014 Paper 2 Q10 (with bearings) • 2018 Paper 2 Q13 (with bearings) • 2019 Paper 2 Q7 • 2022 Paper 1 Q9 • 2024 Paper 2 Q3 |
| Standard Grade: Credit (1986–2013) • Exam questions and answers • More exam questions and answers |
| Intermediate 2 (2000–2015) • Triangle trigonometry (with answers) |
Buy our favourite N5 textbook
Zeta National 5+ Maths
Clear and comprehensive.
Progressive exercises.
Includes answers.
Buy from Zeta Press
Cosine rule worksheets
| Essential Skills worksheet • Cosine rule (Answers) |
|
CJ Maths worksheet • Cosine rule (no answers) |
|
Maths Genie worksheet • Cosine rule (Solutions) |
|
Corbettmaths worksheet • Sine and cosine rules (Answers) |
| Larkhall Academy exercises • Pages 6-9 Ex 4-6 (no answers) |
Buy N5 Maths revision guides
How to Pass N5 Maths TOP CHOICEBrightRED: N5 Maths Study Guide
CGP: N5 Maths Revision Guide
Notes and videos
| Videos - Maths180.com |
| Videos - Mr Graham Maths 1. Finding a side 2. Finding an angle |
| Videos - Corbett Maths 1. Finding a side 2. Finding an angle |
| PowerPoint - MathsRevision.com |
| Notes and videos - Mistercorzi |
| Theory guide - National 5 Maths |
| Notes - National5.com |
| Worked examples - Maths Mutt |
| • Revision notes - BBC Bitesize • Test yourself - BBC Bitesize |
| Notes - Maths4Scotland |
| Notes - D R Turnbull |
Click here to study the cosine rule notes on National5.com.
You may also want to check out their Nat 5 Maths self-study course. We think it's excellent. If you decide to enrol, just use coupon code "Maths.scot" to get £10 discount.
|
⇦ Nat 5 topic list ⇧ Top of this page
|