Answer to Question #348766 in Complex Analysis for Busi

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
"r^2=x^2+y^2=(\\sqrt{3})^2+(1)^2=4""\\tan\\theta=\\dfrac{y}{x}=\\dfrac{1}{\\sqrt{3}}"

Quadrant I

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




"r^2=x^2+y^2=(1)^2+(-1)^2=2""\\tan\\theta=\\dfrac{-1}{1}=-1"

Quadrant IV

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



"z_1\/z_2"


"\\theta_1-\\theta_2=\\dfrac{\\pi}{6}-\\dfrac{7\\pi}{4}=\\dfrac{-19\\pi}{12}"

"Arg(z_1\/z_2)=\\dfrac{-19\\pi}{12}+2\\pi=\\dfrac{5\\pi}{12}"


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