Answer to Question #349054 in Complex Analysis for Mapula Advice

Question

Convert 1−i/2 into trigonometric form

Answer 1

$$r^2=x^2+y^2=(1)^2+(-\dfrac{1}{2})^2=\dfrac{5}{4}$$

$$\tan \theta=\dfrac{-1/2}{1}=-\dfrac{1}{2}$$

Quadrant IV

$$r=\dfrac{\sqrt{5}}{2}, \theta=2\pi-\tan^{-1}\dfrac{1}{2}$$

$$z=\dfrac{\sqrt{5}}{2}(\cos(2\pi-\tan^{-1}\dfrac{1}{2})+i\sin(2\pi-\tan^{-1}\dfrac{1}{2}) )$$

Or

$$r=-\dfrac{\sqrt{5}}{2}, \theta=\pi-\tan^{-1}\dfrac{1}{2}$$

$$z=-\dfrac{\sqrt{5}}{2}(\cos(\pi-\tan^{-1}\dfrac{1}{2})+i\sin(\pi-\tan^{-1}\dfrac{1}{2}) )$$