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The cosine rule

The cosine rule

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Summary

The cosine rule

In a nutshell

The cosine rule can be used to find a missing side or angle of any triangle. There are two forms of the cosine rule that you will need to know.



The cosine rule formula

The cosine rule is used when you have three sides or two sides and the angle between them.


SIDE FORM

ANGLE FORM

​a2=b2+c2−2bccos⁡(A)a^2=b^2+c^2-2bc\cos(A)a2=b2+c2−2bccos(A)​​

​cos⁡(A)=b2+c2−a22bc\cos(A)=\dfrac{b^2+c^2-a^2}{2bc}cos(A)=2bcb2+c2−a2​​​


This is for a triangle with sides a,b,ca,b,ca,b,c and corresponding angles A,B,CA,B,CA,B,C.


Maths; Trigonometry; KS5 Year 12; The cosine rule


Note: It is possible to only memorise one of the formulas and rearrange to find the other.


Example 1

The triangle in the diagram below has side lengths of 5cm5cm5cm​, 7cm7cm7cm​, and 10cm10cm10cm​. What is the size of the angle ∠CAB\angle CAB∠CAB to 222 decimal places?


Maths; Trigonometry; KS5 Year 12; The cosine rule


To find the angle use the angle form of cosine rule:

​cos⁡(A)=b2+c2−a22bc\cos(A)=\dfrac{b^2+c^2-a^2}{2bc}cos(A)=2bcb2+c2−a2​​​


Substitute in known values:

​cos⁡(A)=102+52−722(5)(10)\cos(A)=\dfrac{10^2+5^2-7^2}{2(5)(10)}cos(A)=2(5)(10)102+52−72​​​


​cos⁡(A)=0.76\cos(A)=0.76cos(A)=0.76​​


Use the inverse cosine function to find the angle:

​A=cos⁡−1(0.76)=40.535802...A=\cos^{-1}(0.76)=40.535802...A=cos−1(0.76)=40.535802...​​


​∠CAB=40.54∘ (2 d.p.)‾\underline{\angle CAB=40.54^\circ\space (2\space d.p.)}∠CAB=40.54∘ (2 d.p.)​


Example 2

In the diagram below, calculate the length of the side ccc.​

Maths; Trigonometry; KS5 Year 12; The cosine rule


Note: Sometimes the labelling of a diagram will not use the same notation as the formula.


Write down the side form of the cosine rule:

a2=b2+c2−2bccos⁡(A)a^2=b^2+c^2-2bc\cos(A)a2=b2+c2−2bccos(A)​​


In the formula the missing side length and the angle should be corresponding. Identify each variable:

A=24∘a=cb=2cmc=4cm\begin{aligned}A&= 24^{\circ} \\ a &= c \\ b &=2cm \\ c &=4cm\end{aligned}Aabc​=24∘=c=2cm=4cm​​​


Substitute the values into the formula:

c2=22+42−2(2)(4)(cos⁡(24))c^2 = 2^2 + 4^2 - 2(2)(4)(\cos(24))c2=22+42−2(2)(4)(cos(24))​​


Solve for ccc:

c2=20−16cos⁡(24)c=20−16cos⁡(24)c=2.320188069...\begin{aligned} c^2 &= 20 - 16\cos(24) \\ c &= \sqrt{20-16\cos(24)} \\ c &=2.320188069... \end{aligned}c2cc​=20−16cos(24)=20−16cos(24)​=2.320188069...​​​

c=2.32 cm (2 d.p.)‾\underline {c= 2.32 \ cm\space(2\space d.p.)}c=2.32 cm (2 d.p.)​​​



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Length:
Finding missing lengths in a triangle

Unit 1

Finding missing lengths in a triangle

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The cosine rule

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FAQs - Frequently Asked Questions

What is the side form of the cosine rule?

The side form of the cosine rule is: a^2=b^2+c^2-2bcCos(A).

What is the angle form of the cosine rule?

The angle form of the cosine rule is cos(A)=(b^2+c^2-a^2)/(2bc).

Why is the cosine rule used?

The cosine rule can be used to find a missing side or angle of any triangle.

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