Maths - Cosine Rule | Olly Britton

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

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

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

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

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

When you have SAS.

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

When you have only lengths.

What is the initial setup for the cosine rule?

PHOTO COSINE RULE PROOF UNLABELLED

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.

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

$x$

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

PHOTO COSINE RULE H

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

PHOTO COSINE RULE X

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

PHOTO COSINE RULE XCX

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

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

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

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

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

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

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

\[b\cos A\]

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

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

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

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

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

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

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

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

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

Eliminate $x^2$ from both sides.

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

The cosine rule.

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