How do you apply the sum and difference formula to solve trigonometric equations?

Apr 7, 2015

Main Sum and Differences Trigonometric Identities

$\cos \left(a - b\right) = \cos a \cdot \cos b + \sin a \cdot \sin b$
$\cos \left(a + b\right) = \cos a \cdot \cos b - \sin a \cdot \sin b$
$\sin \left(a - b\right) = \sin a \cdot \cos b - \sin b \cdot \cos a$
$\sin \left(a + b\right) = \sin a \cdot \cos b + \sin b \cdot \cos a$
$\tan \left(a - b\right) = \frac{\tan a - \tan b}{1 + \tan a \cdot \tan b}$
$\tan \left(a + b\right) = \frac{\tan a + \tan b}{1 - \tan a \cdot \tan b}$

Application of Sum and Differences Trigonometric Identities

Example 1: Find $\sin 2 a$.

$\sin 2 a$
$= \sin \left(a + a\right)$
$= \sin a \cdot \cos a + \sin a \cdot \cos a$
$= 2 \cdot \sin a \cdot \cos a$

Example 2: Find $\cos 2 a$.

$\cos 2 a$
$= \cos \left(a + a\right)$
$= \cos a \cdot \cos a - \sin a \cdot \sin a$
$= {\cos}^{2} a - {\sin}^{2} a$

Example 3: Find $\cos \left(\frac{13 \pi}{12}\right)$.

$\cos \left(\frac{13 \pi}{12}\right)$
$= \cos \left(\frac{\pi}{3} + \frac{3 \pi}{4}\right)$
$= \cos \left(\frac{\pi}{3}\right) \cdot \cos \left(\frac{3 \pi}{4}\right) - \sin \left(\frac{\pi}{3}\right) \cdot \sin \left(\frac{3 \pi}{4}\right)$
$= - \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}$
$= - \frac{\sqrt{2} + \sqrt{6}}{4}$