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Prove

arg(z conjugate)=-arg(z)

$L e t z = r (\cos θ + i \sin θ) T h e n, r = | z | A n d, \arg z = θ ...... (1) N o w, \bar{z} = r (\cos θ - i \sin θ) = r (\cos (- θ) + i \sin (- θ)) T h u s, \arg \bar{z} = - θ \Rightarrow - \arg \bar{z} = θ ...... (2) F r o m (1) a n d (2), \arg z = - \arg \bar{z} \Rightarrow \arg \bar{z} = - \arg z$

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