Compound Angle Formulae (Edexcel IAL Maths)

Spec point — 2.3

Compound angle formulae

What are the compound angle formulae?

• There are six compound angle formulae (also known as addition formulae), two each for sin, cos and tan:
• For sin the +/- sign on the left-hand side matches the one on the right-hand side

$\mathrm{sin}(A+B)\equiv\mathrm{sin}A\mathrm{cos}B+\mathrm{cos}A\mathrm{sin}B$ $\mathrm{sin}(A-B)\equiv\mathrm{sin}A\mathrm{cos}B-\mathrm{cos}A\mathrm{sin}B$

• For cos the +/- sign on the left-hand side is opposite to the one on the right-hand side

$\mathrm{cos}(A+B)\equiv\mathrm{cos}A\mathrm{cos}B-\mathrm{sin}A\mathrm{sin}B$ $\mathrm{cos}(A-B)\equiv\mathrm{cos}A\mathrm{cos}B+\mathrm{sin}A\mathrm{sin}B$

• For tan the +/- sign on the left-hand side matches the one in the numerator on the right-hand side, and is opposite to the one in the denominator

$\mathrm{tan}(A+B)\equiv\frac{\mathrm{tan}A+\mathrm{tan}B}{1-\mathrm{tan}A\mathrm{tan}B}$ $\mathrm{tan}(A-B)\equiv\frac{\mathrm{tan}A-\mathrm{tan}B}{1+\mathrm{tan}A\mathrm{tan}B}$

• You can derive the tan identity by:

  • Writing $\mathrm{tan}(A+B)\equiv\frac{\mathrm{sin}(A+B)}{\mathrm{cos}(A+B)}$
  • Dividing the numerator and denominator by $\mathrm{cos}A\mathrm{cos}B$

Exam Hint

• All these formulae are in the formulae booklet – you don't have to memorise them.

Worked Example