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Geometrical Representation of a Complex Number

If argz < 0...

Question

If $a r g (z) < 0$ , then $a r g (- z) - a r g (z) =$

π

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- π

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\frac{π}{2}

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- \frac{π}{2}

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Solution

The correct option is A $π$
Let $Z = r e^{i θ_{1}}$
$- Z = - r e^{i θ_{1}}$
$⟹ - a cos θ_{1} - i b sin θ_{1}$
$⟹ - a cos (π + θ_{1}) - i b sin (π + θ_{1})$
$⟹ r e^{i (π + θ_{1})}$
$a r g (- Z) - a r g (Z)$
$⟹ π + θ_{1} - θ_{1} ⟹ π$

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