Trigonometry / Unit 6: Trigonometric Identities
6.4 Sum and Difference Formulas
Notes
These formulas find trig values of angles like $75^\circ$ and $15^\circ$.
- $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$.
- $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ (note the opposite sign).
- $\displaystyle \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$.
- Example: $\displaystyle \sin 75^\circ = \sin(45^\circ + 30^\circ) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}$.
Practice
What is the exact value of $\cos 75^\circ$?
$\displaystyle \cos(45^\circ + 30^\circ) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$.
What is the value of $\sin 40^\circ \cos 20^\circ + \cos 40^\circ \sin 20^\circ$?
It matches $\sin(A + B)$ with $A = 40^\circ$, $B = 20^\circ$: $\displaystyle \sin 60^\circ = \frac{\sqrt{3}}{2}$.
What is the exact value of $\tan 75^\circ$?
$\displaystyle \tan(45^\circ + 30^\circ) = \frac{1 + \frac{\sqrt{3}}{3}}{1 - \frac{\sqrt{3}}{3}} = \frac{3 + \sqrt{3}}{3 - \sqrt{3}} = 2 + \sqrt{3}$.
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