7.1PureStretch

The Addition Formulae

The addition (compound angle) formulae express trig functions of a sum or difference of angles. They are the foundation for double-angle formulae, the R-form, and many proofs.

28 min Video by Zeeshan Zamurred Trigonometry and Modelling

Edexcel A level Maths: 7.1 The Addition FormulaeWatch the full walkthrough before the notes below.

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What you'll be able to do

State the sin, cos and tan addition formulae
Use them to find exact values
Handle both + and − versions
Recognise the sign patterns

The formulae

For sine the signs match; for cosine they are opposite; tangent has its own form.

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

Sine: signs match.

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

Cosine: signs flip.

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

Tangent.

Finding exact values

Split an awkward angle into a sum or difference of standard angles, e.g. $75° = 45° + 30°$ .

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

= \frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2}

= \frac{6 + 2}{4}

Answer

\frac{6 + 2}{4}

Tip — For cos, remember the sign flips: cos(A+B) uses a minus.

Formula recap

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

Sine addition.

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

Cosine addition.

tan (A \pm B) = \frac{t a n A \pm t a n B}{1 \mp t a n A t a n B}

Tangent addition.

Common mistakes to avoid

Using a plus inside cos(A+B) expansion.

cos(A+B) = cosAcosB − sinAsinB — the sign flips.

Thinking sin(A+B) = sinA + sinB.

It is sinAcosB + cosAsinB.

Key takeaways

sin(A±B) = sinAcosB ± cosAsinB.
cos(A±B) = cosAcosB ∓ sinAsinB.
tan(A±B) = (tanA ± tanB)/(1 ∓ tanAtanB).

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