Course 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).

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:
• We are given two sides and the included angle. We have to find the other side.
• We are given all three sides and have to find one of the angles.
• 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.

Textbook page references

• TeeJay Maths Book N5 pp.81-84
• Zeta National 5+ Maths pp.228-230
• Leckie National 5 Maths pp.302-305

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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.

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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.

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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.

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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.

N5 Maths practice papers

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.

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Past paper questions

Other great resources

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

Lesson notes - Maths 777 1. Finding a length 2. Finding an angle 3. Mixed sine and cosine rules

• Revision notes - BBC Bitesize • Test yourself - BBC Bitesize

Notes - Maths4Scotland

Notes - D R Turnbull

Essential Skills worksheet • Cosine rule worksheet (Answers)

Exercises - Larkhall Academy Pages 6-9 Ex 4-6