Compound-angle formulae

The expansions of sin, cos and tan of α ± β, how they are proved from cos(α − β), and using them for exact values and simplifying expressions.

The six formulae

Compound angles

$$\begin{aligned}\sin(\alpha + \beta) &= \sin\alpha\cos\beta + \cos\alpha\sin\beta \\ \sin(\alpha - \beta) &= \sin\alpha\cos\beta - \cos\alpha\sin\beta \\ \cos(\alpha + \beta) &= \cos\alpha\cos\beta - \sin\alpha\sin\beta \\ \cos(\alpha - \beta) &= \cos\alpha\cos\beta + \sin\alpha\sin\beta \\ \tan(\alpha + \beta) &= \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta} \\ \tan(\alpha - \beta) &= \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta}\end{aligned}$$

Where they come from

y = cos(x − β) and its expansion

Orange: y = cos(x − β). Blue dashed: y = cos x cos β + sin x sin β. They lie exactly on top of each other for every β.

Try this

Set $\beta = \dfrac{\pi}{2}$. Which simpler identity does the expansion become?

Exact values

Example 1●●

Find the exact value of $\sin 75^\circ$.

$\sin 75^\circ = \sin(45^\circ + 30^\circ) = \sin 45^\circ\cos 30^\circ + \cos 45^\circ\sin 30^\circ$

$= \dfrac{1}{\sqrt2} \times \dfrac{\sqrt3}{2} + \dfrac{1}{\sqrt2} \times \dfrac12 = \dfrac{\sqrt3 + 1}{2\sqrt2} = \dfrac{\sqrt6 + \sqrt2}{4}$

Example 2●●

Find the exact value of $\tan 15^\circ$.

$\tan(45^\circ - 30^\circ) = \dfrac{1 - \frac{1}{\sqrt3}}{1 + \frac{1}{\sqrt3}} = \dfrac{\sqrt3 - 1}{\sqrt3 + 1}$

$= \dfrac{(\sqrt3 - 1)^2}{2} = \dfrac{4 - 2\sqrt3}{2} = 2 - \sqrt3$

Multiply top and bottom by √3 − 1.

Example 3: using given ratios●●●

$A$ and $B$ are acute, $\sin A = \dfrac35$ and $\cos B = \dfrac{5}{13}$. Find $\cos(A + B)$.

$\cos A = \dfrac45$ and $\sin B = \dfrac{12}{13}$.

Draw right triangles: 3-4-5 and 5-12-13.

$\cos(A + B) = \dfrac45 \times \dfrac{5}{13} - \dfrac35 \times \dfrac{12}{13} = \dfrac{20 - 36}{65} = -\dfrac{16}{65}$

Collapsing an expansion

Example 4●●

Simplify $\sin 3x\cos x + \cos 3x\sin x$ and $\cos 50^\circ\cos 10^\circ - \sin 50^\circ\sin 10^\circ$.

The first is the expansion of $\sin(3x + x) = \sin 4x$.

The second is $\cos(50^\circ + 10^\circ) = \cos 60^\circ = \tfrac12$.

Flashcards

$sin(lpha + eta)$

$\sin\alpha\cos\beta + \cos\alpha\sin\beta$

$cos(lpha + eta)$

$\cos\alpha\cos\beta - \sin\alpha\sin\beta$

$cos(lpha - eta)$

$\cos\alpha\cos\beta + \sin\alpha\sin\beta$

$ an(lpha + eta)$

$\dfrac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}$

$sin 75^circ$

$\dfrac{\sqrt6 + \sqrt2}{4}$

Practice

$\cos(\alpha + \beta) = $
$\sin(\alpha - \beta) = $
Find the exact value of $\sin 75^\circ$. Type it like (sqrt6+sqrt2)/4.
Find the exact value of $\tan 15^\circ$.
$A$, $B$ acute, $\sin A = \dfrac35$, $\cos B = \dfrac{5}{13}$. Find $\cos(A + B)$.
With the same $A$ and $B$, find $\sin(A + B)$.
Simplify $\sin 3x\cos x + \cos 3x\sin x$.
Evaluate $\cos 50^\circ\cos 10^\circ - \sin 50^\circ\sin 10^\circ$.
$\tan(\alpha - \beta) = $
Expanding $\cos(x - 90^\circ)$ gives: