Question

Express the following in the form A + iB :
$\frac{1}{1-\cos \theta +2i\sin \theta }$

Answer 1

$\frac{1}{1-cos\theta +2isin\theta }=\frac{1}{2si{n}^2\frac{\theta }{2}+4isin\frac{\theta }{2}cos\frac{\theta }{2}}$

$=\frac{1}{2sin\left(\theta /2\right)}\cdot \frac{1}{\left[sin\right(\theta /2)+2icos(\theta /2\left)\right]}$

$=\frac{1}{2sin\left(\theta /2\right)}\cdot \frac{sin\left(\theta /2\right)-2icos\left(\theta /2\right)}{si{n}^2\left(\theta /2\right)+4co{s}^2\left(\theta /2\right)}$

$=\frac{1-2icot\left(\theta /2\right)}{2si{n}^2\left(\theta /2\right)+4\left[2co{s}^2\right(\theta /2\left)\right]}$

Change the double angle.

$=\frac{1-2icot\left(\theta /2\right)}{5+3cos\theta }$