# How do you use half angle identities to solve equations?

Apr 12, 2015

Common Half angle identity:
1. $\sin a = 2 \sin \left(\frac{a}{2}\right) \cdot \cos \left(\frac{a}{2}\right)$

Half angle Identities in term of t = tan a/2.
2. $\sin a = \frac{2 t}{1 + {t}^{2}}$

3.$\cos a = \frac{1 - {t}^{2}}{1 + {t}^{2}}$

1. $\tan a = \frac{2 t}{1 - {t}^{2}} .$

Use of half angle identities to solve trig equations.

Example. Solve $\cos x + 2 \cdot \sin x = 1 + \tan \left(\frac{x}{2}\right) .$
Solution. Call $t = \tan \left(\frac{x}{2}\right)$. Use half angle identities (2) and (3) to transform the equation.

$\frac{1 - {t}^{2}}{4} + \frac{1 + {t}^{2}}{4} = 1 + t .$

$1 - {t}^{2} + 4 t = \left(1 + t\right) \left(1 + {t}^{2}\right)$

${t}^{3} + 2 {t}^{2} - 3 t = t \cdot \left({t}^{2} + 2 t - 3\right) = 0.$

Next, solve the $3$ basic trig equations: tan (x/2) = t = 0; tan (x/2) = -3; and $\tan \left(\frac{x}{2}\right) = 1.$