What Are Compound Angle Identities?

Compound Angle Identity: A trigonometric formula that expresses the sine, cosine, or tangent of the sum or difference of two angles in terms of the trigonometric functions of the individual angles.

Compound angle identities are one of the most powerful tools in your trigonometry toolkit. They allow you to evaluate trig functions at non-standard angles by breaking them into combinations of familiar ones , and they form the foundation from which double angle identities (and even triple angle identities, via de Moivre's theorem) are derived.

There are three families to know:

Sine compound angle identities
Cosine compound angle identities
Tangent compound angle identities

All three appear on the IB Formula Booklet, but understanding where they come from and how to use them fluently is what separates exam-ready students from those who freeze mid-question.

Exam Tip

These identities are provided in the IB Formula Booklet, so you don't need to memorise them verbatim , but you absolutely must know how to apply them confidently and quickly under timed conditions.

Sine Compound Angle Identity

The sine compound angle identity handles the sine of a sum or difference of two angles $α$ and $β$ :

$sin (α \pm β) = sin α cos β \pm cos α sin β$

Notice that the sign on the right matches the sign on the left , a sum gives a sum, a difference gives a difference. This makes the sine version the most intuitive of the three.

Analogy

Think of this identity as a "cross-multiplication" of sine and cosine: the first angle contributes its sine, the second its cosine , and then they swap. The sign tracks through unchanged.

Example

Calculate $sin (75°)$ exactly using the compound angle identity.

We write $75° = 45° + 30°$ , since both are standard angles we know exactly.

$sin (75°) = sin (45° + 30°) = sin 45° cos 30° + cos 45° sin 30°$

Substituting exact values:
$= \frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2} = \frac{6}{4} + \frac{2}{4} = \frac{6 + 2}{4}$

You can verify this numerically: $\frac{6 + 2}{4} \approx 0.9659 = sin 75°$ ✓

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What Are Compound Angle Identities?

There are three families to know:

Sine compound angle identities
Cosine compound angle identities
Tangent compound angle identities

All three appear on the IB Formula Booklet, but understanding where they come from and how to use them fluently is what separates exam-ready students from those who freeze mid-question.

Exam Tip

These identities are provided in the IB Formula Booklet, so you don't need to memorise them verbatim , but you absolutely must know how to apply them confidently and quickly under timed conditions.

Sine Compound Angle Identity

The sine compound angle identity handles the sine of a sum or difference of two angles $α$ and $β$ :

$sin (α \pm β) = sin α cos β \pm cos α sin β$

Notice that the sign on the right matches the sign on the left , a sum gives a sum, a difference gives a difference. This makes the sine version the most intuitive of the three.

Analogy

Think of this identity as a "cross-multiplication" of sine and cosine: the first angle contributes its sine, the second its cosine , and then they swap. The sign tracks through unchanged.

Example

Calculate $sin (75°)$ exactly using the compound angle identity.

We write $75° = 45° + 30°$ , since both are standard angles we know exactly.

$sin (75°) = sin (45° + 30°) = sin 45° cos 30° + cos 45° sin 30°$

Substituting exact values:
$= \frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2} = \frac{6}{4} + \frac{2}{4} = \frac{6 + 2}{4}$

You can verify this numerically: $\frac{6 + 2}{4} \approx 0.9659 = sin 75°$ ✓

Free preview

AHL 3.10—Compound angle identities

What Are Compound Angle Identities?

Sine Compound Angle Identity

9 more sections in this topic

AHL 3.10—Compound angle identities

What Are Compound Angle Identities?

Sine Compound Angle Identity

9 more sections in this topic