National 5MathsSQARevision NotesTrigonometric SkillsTrigonometry with TrianglesCosine Rule

Cosine Rule (SQA National 5 Maths): Revision Note

Exam code: X847 75

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

Using cosine rule to find a length

What is the cosine rule?

  • The cosine rule is used in non right-angled triangles

    • It allows us to find missing side lengths or angles

  • It states that for any triangle

a squared equals b squared plus c squared minus 2 b c cos A

or

cos A equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction

  • Where

    • a is the side opposite angle A

    • b and c are the other two sides

      • b and c are either side of angle A

      • A is the angle between them

    • The second form of the equation is just a rearrangement of the first one

Non Right-Angled Triangle labelled with angles A, B and C and opposite corresponding sides a, b and c.

Examiner Tips and Tricks

Both forms of the cosine rule equation are given to you on the Formulae List in the exam paper.

How do I use the cosine rule to find a missing length?

  • Use the cosine rule for lengths

    • when you have two sides and the angle between them

    • and you want to find the opposite side

  • Start by labelling your triangle with the angles and sides

    • Angles have upper case letters

    • Sides opposite the angles have the equivalent lower case letter

  • Substitute values into the  a squared equals b squared plus c squared minus 2 b c cos A space form of the equation

    • Take the square root to find a

Examiner Tips and Tricks

Swapping the letters around allows the cosine rule equation to be written in these forms as well:

b squared equals a squared plus c squared minus 2 a c cos B space

c squared equals a squared plus b squared minus 2 a b cos C space

Just make sure the lower case letter on the left matches matches the upper case letter in the cosine on the right.

Worked Example

In triangle ABC: 

  • Angle ABC = 109°

  • AB = 8.1 cm

  • BC = 12.3 cm

Triangle ABC with AB = 8.1 cm, BC = 12.3 cm, and angle ABC = 109º.

Calculate the length of AC.

Answer:

Label the sides of the triangle

Triangle ABC with sides opposite angles labelled with corresponding lowercase letters.

You want to find the length of b

  • So swap the letters around in the Formulae List form of the Cosine Rule equation to get b squared equals a squared plus c squared minus 2 a c cos B space

Substitute the known values into that equation

b squared equals 8.1 squared plus 12.3 squared minus 2 cross times 8.1 cross times 12.3 cross times cos 109 space

Take the square root of both sides to find b, and use your calculator to find the value

table row b equals cell square root of 8.1 squared plus 12.3 squared minus 2 cross times 8.1 cross times 12.3 cross times cos 109 space end root end cell row blank equals cell 16.786086... end cell end table

Round to a sensible degree of accuracy

  • Unless a question tells you otherwise, 3 significant figures is usually a good choice

AC = 16.8 cm  (3 s.f.)

Using cosine rule to find an angle

How do I use the cosine rule to find a missing angle?

  • Use the cosine rule for angles

    • when you have all three sides

    • and you want to find an angle

  • Use the cos A equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction form of the equation to find the unknown angle A

    • Remember, A is the angle between sides b and c

      • (you may need to relabel the triangle)

    • You will need to use inverse cosine on your calculator, cos to the power of negative 1 end exponent open parentheses... close parentheses

  • Unlike the sine rule, there is no ambiguous case of the cosine rule

    • The answer given by cos to the power of negative 1 end exponent open parentheses... close parentheses is the only possible angle for the triangle in question

Examiner Tips and Tricks

Swapping the letters around allows the cosine rule equation to be written in these forms as well:

cos B equals fraction numerator a squared plus c squared minus b squared over denominator 2 a c end fraction

cos C equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction

Note that the lower case version of the angle letter on the left only appears after the minus sign in the numerator on the right.

Worked Example

In triangle ABC:

  • AB equals 4.2 space straight m

  • BC equals 7.1 space straight m

  • AC equals 3.8 space straight m.

Triangle ABC with AB = 4.2 m, BC = 7.1 m, AC = 3.8 m. Angle BAC is shaded.

Calculate the size of the shaded angle at straight A.

Answer:

Label the sides of the triangle

Triangle ABC with sides opposite angles labelled with corresponding lowercase letters.

You want to find the size of the angle at A

  • So use the equation cos A equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction from the Formulae List

Substitute the known values into that equation

cos A equals fraction numerator 3.8 squared plus 4.2 squared minus 7.1 squared over denominator 2 cross times 3.8 cross times 4.2 end fraction

Use cos-1 on your calculator to find the value of A

table row A equals cell cos to the power of negative 1 end exponent open parentheses fraction numerator 3.8 squared plus 4.2 squared minus 7.1 squared over denominator 2 cross times 3.8 cross times 4.2 end fraction close parentheses end cell row blank equals cell 125.046994... end cell end table

Round to a sensible degree of accuracy

  • Unless a question tells you otherwise, 1 decimal place is usually a good choice for angles

A equals 125.0 degree space space open parentheses 1 space straight d. straight p. close parentheses

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.