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Q)

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

This the third part of the multi-part Q6.

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A)
• 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).
Let $-1-i=r(\cos \theta+i\sin \theta)$
Therefore $r\cos\theta=-1$ and $r\sin \theta=-1$
$r^2=1^2+1^2$
$r=\sqrt{1+1}=\sqrt 2$
$\alpha=\tan^{-1}\mid\large\frac{-1}{-1}\mid=$$\tan^{-1}1=\large\frac{\pi}{4} Step 2: The point representing -1-i lies in the quadrant 3 Therefore \theta=\pi+\alpha=\pi+\large\frac{\pi}{4}=\frac{5\pi}{4} -1-i=1(\cos\big(\large\frac{5\pi}{4}\big)$$+i\sin\big(\large\frac{5\pi}{4}\big))$