Maths - Cosine Rule | Olly Britton

The two forms of the cosine rule

What is the $a^2$ form of the cosine rule?

a-squared-form

\[a^2 = b^2 + c^2 - 2bc\cos(A)\]

What is the $\cos(A)$ form of the cosine rule?

cos-a-form

\[\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}\]

When to use the cosine rule

When can you use the cosine rule for working out the length of a side?

use-for-side-sas

When you have SAS.

When can you use to cosine rule for working out an angle?

use-for-angle-sss

When you have only lengths.

Setting up the proof diagram

What is the initial setup for the cosine rule?

proof-initial-setup

PHOTO COSINE RULE PROOF UNLABELLED

How is the start of the cosine rule proof similar to the start of the sine rule proof?

proof-similar-to-sine-rule

You drop a line $h$ down the centre of a triangle.

Along with $h$, what variable do you also introduce when starting the cosine rule proof?

proof-introduces-x

$x$

PHOTO COSINE RULE PROOF UNLABELLED Where does $h$ go for the cosine rule proof?

proof-position-of-h

PHOTO COSINE RULE H

PHOTO COSINE RULE PROOF UNLABELLED Where does $x$ go for the cosine rule proof?

proof-position-of-x

PHOTO COSINE RULE X

PHOTO COSINE RULE PROOF X What else could you label on the above diagram in terms of $x$?

proof-label-c-minus-x

PHOTO COSINE RULE XCX

Pythagorean and trigonometric expressions

PHOTO COSINE RULE PROOF LABELLED How could you write $a^2$?

proof-a-squared-expression

\[(c - x)^2 + h^2 = a^2\]

PHOTO COSINE RULE PROOF LABELLED How could you write $b^2$?

proof-b-squared-expression

\[x^2 + h^2 = b^2\]

PHOTO COSINE RULE PROOF LABELLED How could you write $\cos(A)$?

proof-cos-a-expression

\[\frac{x}{b}\]

PHOTO COSINE RULE PROOF LABELLED If $\cos(A) = \frac{x}{b}$, how could you write $x$?

proof-x-in-terms-of-cos-a

\[b\cos A\]

Combining and simplifying the expressions

Rearrange $b^2 = x^2 + h^2$ so $h^2$ is the subject of the formula?

rearrange-h-squared-from-b

\[h^2 = b^2 - x^2\]

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

rearrange-h-squared-from-a

\[h^2 = a^2 - (c-x)^2\]

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

equate-the-two-h-squared

\[b^2 - x^2 = a^2 - (c-x)^2\]

What is $(c-x)^2$?

expand-c-minus-x-squared

\[c^2 - 2cx + x^2\]

How could you simplify $b^2 - x^2 = a^2 - c^2 + 2cx + x^2$?

eliminate-x-squared

Eliminate $x^2$ from both sides.

Substituting $x = b\cos(A)$ into $b^2 = a^2 - c^2 + 2cx$ gives what formula?

substitute-to-get-cosine-rule

The cosine rule.

Summary of the proof

What are the four main steps for proving the cosine rule?

four-steps-of-proof