Complex NumbersHard

Question

If arg (z) < 0, then arg (- z) - arg (z) equals

Options

A.π
B.- π
C.- π/2
D.π/2

Solution

     
Since, arg (z) < 0
⇒       arg (z) = - θ
⇒       z = r cos (- θ) + i sin (- θ)
        = r( cos θ - i sin θ)
And - 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
Alternate Solution
Reason : arg (- z) - arg z
= arg = arg (- 1) = π
and also       arg z - arg (-z )
= arg arg (- 1) = π

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