5.3 Sum and Difference Identities

Notes

• $\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$.
• $\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$ (the sign flips).
• $\displaystyle \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$.
• Use them to get exact values for angles like $75^\circ$, $15^\circ$ and $105^\circ$.
• Example: $\displaystyle \sin 75^\circ = \sin(45^\circ + 30^\circ) = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2}\cdot\frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}$.