Addition formulae

The compound-angle formulae for sin(A±B), cos(A±B) and tan(A±B) — the sign patterns to get right, and using them for exact values of non-standard angles, expanding, proving identities and solving equations.

Pure · Trigonometry

Addition formulae

Why it works

The trig functions are not linear —

\sin(A+B)

is not

\sin A + \sin B

. Instead there are exact "compound angle" formulae (they're in the formula booklet, but knowing them cold is what makes the questions fast):

\sin(A \pm B) \equiv \sin A\cos B \pm \cos A\sin B,

\cos(A \pm B) \equiv \cos A\cos B \mp \sin A\sin B,

\tan(A \pm B) \equiv \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}.

The sign traps — read them carefully:

$\sin$ keeps the sign: $\sin(A-B)$ uses a minus in the middle.
$\cos$ flips the sign: $\cos(A+B)$ uses a minus, $\cos(A-B)$ a plus

(the opposite of the bracket).

$\tan$ has the $\mp$ in the denominator (opposite to the top): for

\tan(A+B)

the bottom is

1 - \tan A\tan B

.

Where they're used.

Exact values of awkward angles — split into two special angles, e.g.

75^\circ = 45^\circ + 30^\circ

15^\circ = 45^\circ - 30^\circ

Expanding $\sin(x + 60^\circ)$ etc. into $a\sin x + b\cos x$ form.
Proving identities — adding or subtracting two expansions makes terms cancel.
Solving — expand a compound angle, then collect into a single $\tan x$ or a

single function.