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# Express the following complex numbers in polar form. $1 - \mathit{i}$

This is the fourth part of the multi-part Q6.

Toolbox:
• If $z=x+iy$ is written in exponential form as $z=r(\cos \theta+i\sin \theta),r=\sqrt{x^2+y^2}$ and the argument $\theta$ is given by the following rule
• $\theta=\pi-\alpha\Rightarrow \theta=\alpha$
• $\theta=-\pi+\alpha\Rightarrow \theta=-\alpha$
• Where $\alpha=\tan^{-1}\mid\large\frac{y}{x}\mid$ and $(x,y)$ lies in one of the four quadrants (or the axes).
Step 1:
Let $1-i=r(\cos \theta+i\sin \theta)$
$r\cos\theta=1,r\sin \theta=-1$
$r^2=1^2+1^2\Rightarrow r=\sqrt{1+1}=\sqrt 2$
$\alpha=\tan^{-1}=\mid \large\frac{-1}{1}\mid=$$\tan^{-1}1 \alpha\Rightarrow \large\frac{\pi}{4} Step 2: The point representing 1-i lies in quadrant 4. Therefore \theta=-\alpha=-\large\frac{\pi}{4} 1-i=i(\cos\big(\large\frac{-\pi}{4}\big)$$+i\sin\big(\large\frac{-\pi}{4}\big))$
edited Jul 19, 2013