First slide

Algebra of complex numbers

Question

z is a unimodular complex number
Statement 1: $\arg (z^{2} + \bar{z}) = \arg z$

Statement 2: $\bar{z} = \cos (\arg z) - i \sin (\arg z)$

Easy

A

Statement -1 is true, statement-2 is true and statement-2 is correct explanation for statement-1
B

Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1
C

Statement-1 is true, statement-2 is false
D

Statement -1 is false, statement-2 is true.

Solution

$| z | = 1 \Rightarrow z = \cos θ + i \sin θ$ where $θ = \arg z$
$\bar{z} = \cos θ - i \sin θ$
Statement -2 is true
$\arg (z^{2} + \bar{z}) = \arg {\cos 2 θ + i \sin 2 θ + \cos θ - i \sin θ}$
$= \tan^{- 1} \frac{\sin 2 θ - \sin θ}{\cos 2 θ + \cos θ}$
$= \tan^{- 1} \frac{2 \cos \frac{3 θ}{2} \sin \frac{θ}{2}}{2 \cos \frac{3 θ}{2} \cos \frac{θ}{2}} = \tan^{- 1} \tan \frac{θ}{2} = \frac{θ}{2} = \frac{1}{2} \arg z$

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