If arg ( z ) 0 , then arg ( - z ) - arg ( z ) = [2000]

Question

If $\arg (z) < 0$ , then $\arg (- z) - \arg (z) =$ [2000]

Smart Achievers · Accepted Answer

(1)

$Given: \arg (z) < 0 (given) \Rightarrow \arg (z) = - θ$

$Now, z = r \cos (- θ) + i \sin (- θ) = r [\cos θ - i \sin θ]$

$Again, - z = - r [\cos θ - i \sin θ] = r [\cos (π - θ) + i \sin (π - θ)]$

$∴$ $\arg (- z) = π - θ$

$Thus \arg (- z) - \arg (z) = (π - θ) - (- θ) = π - θ + θ = π$

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