Compound angle formulae

Covers AS91575

The compound angle formulae

A compound angle is an angle written as a sum or difference of two angles, such as $A + B$ or $A - B$ .
You cannot split the function over the angle: $\sin(A + B) \ne \sin A + \sin B$ .
Instead use the compound angle formulae:

$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$

$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

Reading the sign rules

$\sin$ keeps the same sign: $\sin(A - B) = \sin A \cos B - \cos A \sin B$ .
$\cos$ swaps the sign: $\cos(A + B)$ uses a minus in the middle.
Keep the pattern in order — sin cos, cos sin for sine; cos cos, sin sin for cosine.

Finding exact values

Split an awkward angle into two special angles whose sine and cosine you know exactly ( $30°$ , $45°$ , $60°$ ).
For example $75° = 45° + 30°$ , or $15° = 45° - 30°$ .
Substitute the exact values and simplify.

Using given ratios

You may be given $\sin A$ and $\cos B$ (with the quadrants stated) and asked for $\sin(A + B)$ or $\cos(A - B)$ .
Find the missing ratios first (using $\sin^2\theta + \cos^2\theta = 1$ and the quadrant for the sign), then substitute.

Worked ExampleExact value of $\sin 75°$

Find the exact value of $\sin 75°$ .

Step 1 — Write as a sum of special angles

$75° = 45° + 30°$

Step 2 — Apply the sine formula

$\sin(45° + 30°) = \sin 45° \cos 30° + \cos 45° \sin 30°$

Step 3 — Substitute exact values

$= \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2}\cdot\frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$

Step 4 — Simplify

$\boxed{\sin 75° = \frac{\sqrt{6} + \sqrt{2}}{4}}$

Worked ExampleCompound angle from given ratios

$A$ and $B$ are both acute, with $\sin A = \tfrac{3}{5}$ and $\cos B = \tfrac{5}{13}$ . Find $\cos(A + B)$ .

Step 1 — Find the missing ratios

Both angles are acute, so cosine and sine are positive:

$\cos A = \frac{4}{5}, \qquad \sin B = \frac{12}{13}$

Step 2 — Apply the cosine formula

$\cos(A + B) = \cos A \cos B - \sin A \sin B$

Step 3 — Substitute and simplify

$= \frac{4}{5}\cdot\frac{5}{13} - \frac{3}{5}\cdot\frac{12}{13} = \frac{20}{65} - \frac{36}{65}$

$\boxed{\cos(A + B) = -\frac{16}{65}}$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Find the exact value of $\cos 15°$ using a compound angle formula.

Merit

$A$ is acute with $\sin A = \tfrac{4}{5}$ , and $B$ is obtuse with $\sin B = \tfrac{5}{13}$ . Find the exact value of $\sin(A + B)$ .

Excellence

Prove that $\sin(A + B) + \sin(A - B) = 2\sin A \cos B$ .

Compound angle formulae

Covers AS91575

The compound angle formulae

A compound angle is an angle written as a sum or difference of two angles, such as $A + B$ or $A - B$ .
You cannot split the function over the angle: $\sin(A + B) \ne \sin A + \sin B$ .
Instead use the compound angle formulae:

$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$

$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

Reading the sign rules

$\sin$ keeps the same sign: $\sin(A - B) = \sin A \cos B - \cos A \sin B$ .
$\cos$ swaps the sign: $\cos(A + B)$ uses a minus in the middle.
Keep the pattern in order — sin cos, cos sin for sine; cos cos, sin sin for cosine.

Finding exact values

Split an awkward angle into two special angles whose sine and cosine you know exactly ( $30°$ , $45°$ , $60°$ ).
For example $75° = 45° + 30°$ , or $15° = 45° - 30°$ .
Substitute the exact values and simplify.

Using given ratios

You may be given $\sin A$ and $\cos B$ (with the quadrants stated) and asked for $\sin(A + B)$ or $\cos(A - B)$ .
Find the missing ratios first (using $\sin^2\theta + \cos^2\theta = 1$ and the quadrant for the sign), then substitute.

Worked ExampleExact value of $\sin 75°$

Find the exact value of $\sin 75°$ .

Step 1 — Write as a sum of special angles

$75° = 45° + 30°$

Step 2 — Apply the sine formula

$\sin(45° + 30°) = \sin 45° \cos 30° + \cos 45° \sin 30°$

Step 3 — Substitute exact values

$= \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2}\cdot\frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$

Step 4 — Simplify

$\boxed{\sin 75° = \frac{\sqrt{6} + \sqrt{2}}{4}}$

Worked ExampleCompound angle from given ratios

$A$ and $B$ are both acute, with $\sin A = \tfrac{3}{5}$ and $\cos B = \tfrac{5}{13}$ . Find $\cos(A + B)$ .

Step 1 — Find the missing ratios

Both angles are acute, so cosine and sine are positive:

$\cos A = \frac{4}{5}, \qquad \sin B = \frac{12}{13}$

Step 2 — Apply the cosine formula

$\cos(A + B) = \cos A \cos B - \sin A \sin B$

Step 3 — Substitute and simplify

$= \frac{4}{5}\cdot\frac{5}{13} - \frac{3}{5}\cdot\frac{12}{13} = \frac{20}{65} - \frac{36}{65}$

$\boxed{\cos(A + B) = -\frac{16}{65}}$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Find the exact value of $\cos 15°$ using a compound angle formula.

Merit

$A$ is acute with $\sin A = \tfrac{4}{5}$ , and $B$ is obtuse with $\sin B = \tfrac{5}{13}$ . Find the exact value of $\sin(A + B)$ .

Excellence

Prove that $\sin(A + B) + \sin(A - B) = 2\sin A \cos B$ .