Answer to Question #262542 in Complex Analysis for luka

Question

Determine the modulus and argument of z1=1+cos θ+isin θ/1-cos θ+isin θ

Answer 1

$$\dfrac{1+\cos θ+i\sin θ}{1-\cos θ+i\sin θ}$$

$$=\dfrac{2\cos^2(\dfrac{\theta}{2})+2\sin(\dfrac{\theta}{2})\cos(\dfrac{\theta}{2})i}{2\sin^2(\dfrac{\theta}{2})+2\sin(\dfrac{\theta}{2})\cos(\dfrac{\theta}{2})i}$$

$$=\dfrac{2\cos(\dfrac{\theta}{2})\big(\cos(\dfrac{\theta}{2})+\sin(\dfrac{\theta}{2})i\big)}{2\sin(\dfrac{\theta}{2})\big(\cos(\dfrac{\theta}{2})+\sin(\dfrac{\theta}{2})i\big)}$$

$$=\cot(\dfrac{\theta}{2}), \sin(\dfrac{\theta}{2})\not=0$$

$$|z|=\cot(\dfrac{\theta}{2}), Arg(z)=0, \theta\not=\pi n, n\in\Z$$