First slide

Algebra of complex numbers

Question

If $z = \frac{1 + \cos θ + i \sin θ}{\sin θ + i (1 + \cos θ)} (0 < θ < π / 2)$
then $| z |$ equals

Moderate

A

$2 | \sin θ |$
B

$2 | \cos θ |$
C

1
D

$| \cot (θ / 2) |$

Solution

Using $|\frac{z_{1}}{z_{2}}| = \frac{|z_{1}|}{|z_{2}|} if z_{2} \neq 0$ , we get
$\begin{array}{l} | z |^{2} = \frac{(1 + \cos θ)^{2} + \sin^{2} θ}{\sin^{2} θ + (1 + \cos θ)^{2}} = 1 \\ | z | = 1 \end{array}$

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