Compound Angle Formulae (DP IB Analysis & Approaches (AA): HL): Revision Note

Compound angle formulae

What are the compound angle formulae?

• The compound angle formulae are:

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

    • The sign is the same on both sides

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

    • The sign changes

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

    • The sign is the same on the numerator

    • But changes on the denominator

How can I use the compound angle formulas to find exact trig values?

• STEP 1
  Write the angle as a sum or difference of multiples of 30° or 45°

  • e.g. 15° = 45° - 30°

• STEP 2
  Use the relevant compound angle formula

  • e.g. $\sin \left(45-30\right)=\sin 45\cos 30-\cos 45\sin 30$

• STEP 3
  Evaluate the known trig values and simplify

  • e.g. $\sin 15=\frac{\sqrt{2}}{2}\times \frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\times \frac{1}{2}=\frac{\sqrt{6}-\sqrt{2}}{2}$

How can I use the compound angle formulas?

• You can use the compound angle formulas to prove identities are true

  • You can derive the double angle formulas

• You can use the formulas to help you solve equations