Cosine Rule - Formula, Revision and worksheets. | Maths Made Easy

# Cosine Rule – Formula, Revision and worksheets.

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## What you need to know

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

It is expressed 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

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

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

### Example Questions

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 $b=86$ and $c=65$.

Now, subbing these values into the cosine rule equation, we get

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

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

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

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 is $A=6$. Then, it doesn’t matter how we choose the other two sides, so we will let $b=5$ and $c=7$.

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

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

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

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

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