Deriving the Addition Formulae

Trig Refresher

Deriving Addition Formulae

ABD is a triangle on the unit circle.

Rotate ABD through angle β:

A → A′
D → D′

But AD² = (A′D′)²

So:

Also:

and

Double Angles

Multiple Angles

Example

Show that sin 3A = 3 sin A − 4 sin³A

Half Angles

Example

Show that:

\[ \sin\!\left(\frac{A}{2}\right) = \sqrt{\frac{1 - \cos A}{2}} \]

Start by using the cosine double angle formula:

\[ \cos 2\alpha = \cos^2\alpha - \sin^2\alpha \] \[ = 1 - 2\sin^2\alpha \] \[ = 2\cos^2\alpha - 1 \]

Let α = A/2, so 2α = A.

Substitute:

\[ \cos A = 1 - 2\sin^2\!\left(\frac{A}{2}\right) \] \[ 2\sin^2\!\left(\frac{A}{2}\right) = 1 - \cos A \] \[ \sin^2\!\left(\frac{A}{2}\right) = \frac{1 - \cos A}{2} \] \[ \sin\!\left(\frac{A}{2}\right) = \pm \sqrt{\frac{1 - \cos A}{2}} \]