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.

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

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 $$5cm$$​, $$7cm$$​, and $$10cm$$​. What is the size of the angle $$\angle CAB$$ to $$2$$ decimal places?

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

​$$\cos(A)=\dfrac{b^2+c^2-a^2}{2bc}$$​​

Substitute in known values:

​$$\cos(A)=\dfrac{10^2+5^2-7^2}{2(5)(10)}$$​​

​$$\cos(A)=0.76$$​​

Use the inverse cosine function to find the angle:

​$$A=\cos^{-1}(0.76)=40.535802...$$​​

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

Example 2

In the diagram below, calculate the length of the side $$c$$.​

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:

$$a^2=b^2+c^2-2bc\cos(A)$$​​

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

$$\begin{aligned}A&= 24^{\circ} \\ a &= c \\ b &=2cm \\ c &=4cm\end{aligned}$$​​

Substitute the values into the formula:

$$c^2 = 2^2 + 4^2 - 2(2)(4)(\cos(24))$$​​

Solve for $$c$$:

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

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

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