Question #348766

Write z1=(√3+i) and z2=(1−i) in trigonometric form. Then the argument of the quotient (z1)/(z2)is given by


1
Expert's answer
2022-06-07T17:09:31-0400
r2=x2+y2=(3)2+(1)2=4r^2=x^2+y^2=(\sqrt{3})^2+(1)^2=4tanθ=yx=13\tan\theta=\dfrac{y}{x}=\dfrac{1}{\sqrt{3}}

Quadrant I

z1=3+i=2(cos(π6)+isin(π6))z_1=\sqrt{3}+i=2(\cos(\dfrac{\pi}{6})+i\sin(\dfrac{\pi}{6}))




r2=x2+y2=(1)2+(1)2=2r^2=x^2+y^2=(1)^2+(-1)^2=2tanθ=11=1\tan\theta=\dfrac{-1}{1}=-1

Quadrant IV

z2=1i=2(cos(7π4)+isin(7π4))z_2=1-i=\sqrt{2}(\cos(\dfrac{7\pi}{4})+i\sin(\dfrac{7\pi}{4}))



z1/z2z_1/z_2


θ1θ2=π67π4=19π12\theta_1-\theta_2=\dfrac{\pi}{6}-\dfrac{7\pi}{4}=\dfrac{-19\pi}{12}

Arg(z1/z2)=19π12+2π=5π12Arg(z_1/z_2)=\dfrac{-19\pi}{12}+2\pi=\dfrac{5\pi}{12}


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