# Maths - Cosine Rule

> Source: https://ollybritton.com/notes/a-level/maths/topics/cosine-rule/ · Updated: 2024-08-25 · Tags: maths, school, year-1

### The two forms of the cosine rule
##### What is the $a^2$ form of the cosine rule??
$$
a^2 = b^2 + c^2 - 2bc\cos(A)
$$
^a-squared-form

##### What is the $\cos(A)$ form of the cosine rule??
$$
\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}
$$
^cos-a-form

### When to use the cosine rule
##### When can you use the cosine rule for working out the length of a side??
When you have SAS.
^use-for-side-sas

##### When can you use to cosine rule for working out an angle??
When you have only lengths.
^use-for-angle-sss

### Setting up the proof diagram
##### What is the initial setup for the cosine rule??
![PHOTO COSINE RULE PROOF UNLABELLED](cosine-rule-proof-unlabelled.png)
^proof-initial-setup

##### How is the start of the cosine rule proof similar to the start of the sine rule proof??
You drop a line $h$ down the centre of a triangle.
^proof-similar-to-sine-rule

##### Along with $h$, what variable do you also introduce when starting the cosine rule proof??
$x$
^proof-introduces-x

##### ![PHOTO COSINE RULE PROOF UNLABELLED](cosine-rule-proof-unlabelled.png) Where does $h$ go for the cosine rule proof??
![PHOTO COSINE RULE H](cosine-rule-proof-h.png)
^proof-position-of-h

##### ![PHOTO COSINE RULE PROOF UNLABELLED](cosine-rule-proof-unlabelled.png) Where does $x$ go for the cosine rule proof??
![PHOTO COSINE RULE X](cosine-rule-proof-x.png)
^proof-position-of-x

##### ![PHOTO COSINE RULE PROOF X](cosine-rule-proof-x.png) What else could you label on the above diagram in terms of $x$??
![PHOTO COSINE RULE XCX](cosine-rule-proof-labelled.png)
^proof-label-c-minus-x

### Pythagorean and trigonometric expressions
##### ![PHOTO COSINE RULE PROOF LABELLED](cosine-rule-proof-labelled.png) How could you write $a^2$??
$$
(c - x)^2 + h^2 = a^2
$$
^proof-a-squared-expression

##### ![PHOTO COSINE RULE PROOF LABELLED](cosine-rule-proof-labelled.png) How could you write $b^2$??
$$
x^2 + h^2 = b^2
$$
^proof-b-squared-expression

##### ![PHOTO COSINE RULE PROOF LABELLED](cosine-rule-proof-labelled.png) How could you write $\cos(A)$??
$$
\frac{x}{b}
$$
^proof-cos-a-expression

##### ![PHOTO COSINE RULE PROOF LABELLED](cosine-rule-proof-labelled.png) If $\cos(A) = \frac{x}{b}$, how could you write $x$??
$$
b\cos A
$$
^proof-x-in-terms-of-cos-a

### Combining and simplifying the expressions
##### Rearrange $b^2 = x^2 + h^2$ so $h^2$ is the subject of the formula??
$$
h^2 = b^2 - x^2
$$
^rearrange-h-squared-from-b

##### Rearrange $a^2 = (c-x)^2 + h^2$ so $h^2$ is the subject of the formula??
$$
h^2 = a^2 - (c-x)^2
$$
^rearrange-h-squared-from-a

##### If $h^2 = b^2 - x^2$ and $h^2 = a^2 - (c-x)^2$, then what must be true??
$$
b^2 - x^2 = a^2 - (c-x)^2
$$
^equate-the-two-h-squared

##### What is $(c-x)^2$??
$$
c^2 - 2cx + x^2
$$
^expand-c-minus-x-squared

##### How could you simplify $b^2 - x^2 = a^2 - c^2 + 2cx + x^2$??
Eliminate $x^2$ from both sides.
^eliminate-x-squared

##### Substituting $x = b\cos(A)$ into $b^2 = a^2 - c^2 + 2cx$ gives what formula??
The cosine rule.
^substitute-to-get-cosine-rule

### Summary of the proof
##### What are the four main steps for proving the cosine rule??
* Drawing a diagram with two new variables
* Creating two expressions using the Pythagorean theorem
* Creating one expression using normal trigonometry
* Substituting
^four-steps-of-proof

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