Sum and Difference Of Two Angles (Addition or Compound Angle Formula) - SS3 Mathematics Lesson Note

1. \(\sin(A + B) = \sin A\cos B + \cos A\sin B\)

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

3. \(\sin(A - B) = \sin A\cos B - \cos A\sin B\)

4. \(\cos(A - B) = \cos A\cos B + \sin A\sin B\)

5. \(\tan{(A + B)} = \frac{\sin(A + B)}{\cos(A + B)} = \frac{\tan A + \tan B}{1 - \tan A\tan B}\)

6. \(\tan{(A - B)} = \frac{\sin(A - B)}{\cos(A - B)} = \frac{\tan A - \tan B}{1 + \tan A\tan B}\)

Example: Without using tables, find the value of \(\sin{15{^\circ}}\) in surd form.

Solution

\[\sin{15{^\circ}} = \sin{(45{^\circ} - 30{^\circ})}\]

\[\sin(A - B) = \sin A\cos B - \cos A\sin B\]

\[\sin(45{^\circ} - 30{^\circ}) = \sin{45{^\circ}}\cos{30{^\circ}} - \cos{45{^\circ}}\sin{30{^\circ}}\]

\[= \frac{\sqrt{2}}{2}.\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2}.\frac{1}{2}\]

\[= \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}\]