Cosine Rule

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Cosine Rule Revision

Cosine Rule

When we first learn the cosine function, we learn how to use it to find missing side-lengths & angles in right-angled triangles. The cosine rule is an equation that can help us find missing side-lengths and angles in any triangle.

Make sure you are happy with the following topics before continuing:

– Trigonometry

– Rearranging Formula

Cosine Rule Formula

The cosine rule is an equation that helps us find missing side-lengths and angles in any triangle.

It is expressed according to the triangle on the right.

The cosine rule is

$\textcolor{limegreen}{a}^2=\textcolor{blue}{b}^2+\textcolor{red}{c}^2-2\textcolor{blue}{b}\textcolor{red}{c}\cos \textcolor{limegreen}{A}$

In this topic, we’ll go through examples of cosine rule questions.

Example: Cosine rule to find a length

Use the cosine rule to find the side-length marked $x$ to $1$ dp.

[3 marks]

First we match up the information in the question to the letters in the formula.

$a=x$ , $A = 44\degree$ , $b=5$ and $c=7$ .

So, we’re ready to substitute the values into the formula.

$x^2=5^2+7^2-(2\times5\times7\times\cos(44))=25+49-70\cos(44)$

Taking the square root of both sides, and putting it into the calculator, we get

$x=\sqrt{25+49-70\cos(44)}=4.9$ ( $1$ dp).

Cosine Rule Example Questions

Question 1: Use the cosine rule to find the side-length marked $x$ below to $2$ significant figures.

[3 marks]

Cosine rule to find a length

Firstly, we need appropriately label the sides of this triangle. Firstly, we set $a=x$ , and therefore we get that $A=19$ , since it is the angle opposite. It doesn’t matter how we label the other two sides, so here we’ll let $b=86$ cm and $c=65$ cm. Now, subbing these values into the cosine rule equation, we get

$x^2=86^2+65^2-(2\times86\times65\times\cos(19\degree))=7,396+4,225-11,180\cos(19\degree)$

Then, taking the square root, and putting it into the calculator, we get,

$x=\sqrt{7,396+4,225-11,180\cos(19\degree)}=32$ cm ( $2$ sf).

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Question 2: Use the cosine rule to find the angle marked $x$ below to $1$ decimal place.

[3 marks]

Cosine rule to find an angle

As always, we must label our triangle. Firstly, assign the thing we’re looking for to be $A=x$ , and therefore make the side opposite to it $a=6$ mm. Then, it doesn’t matter how we choose the other two sides, so we will let $b=5$ mm and $c=7$ mm.

Here, we will use the rearranged version of the formula that looks like

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

So, subbing these values into the equation, we get

$\begin{aligned} \cos x &=\dfrac{5^2+7^2-6^2}{2\times5\times7} \\ \\ &=\dfrac{25+49-36}{70}\end{aligned}$

Taking $\cos^{-1}$ of both sides, and putting it into a calculator, we get

$x=\cos^{-1}\left(\dfrac{25+49-36}{70}\right)=57.1\degree$ ( $3$ sf).

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Question 3: Use the cosine rule to find the side-length marked $x$ below to $2$ significant figures.

[3 marks]

Firstly, we need appropriately label the sides of this triangle. Firstly, we set $a=x$ , and therefore we get that $A=40\degree$ , since it is the angle opposite. It doesn’t matter how we label the other two sides, so here we’ll let $b=6$ cm and $c=4.5$ cm. Now, substiuting these values into the cosine rule equation, we get,

$x^2=6^2+4.5^2-(2\times6\times4.5\times\cos(40\degree))=36+20.25-54\cos(40\degree)$

Then, taking the square root, and putting it into the calculator, we get,

$x=\sqrt{56.25-54\cos(40\degree)}=3.9$ cm ( $2$ sf).

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