Answer to Question #349057 in Calculus for Mapula Advice

Question #349057

Write z1=3–√+i and z2=1−i in trigonometric form. Then the argument of the quotient z1/z2is given by

1
Expert's answer
2022-06-13T05:12:03-0400
"r^2=(-\\sqrt{3})^2+(1)^2=4""r\\ge0=>r=\\sqrt{4}=2"

Quadrant II


"\\tan \\theta=\\dfrac{-1}{\\sqrt{3}}=-\\dfrac{1}{\\sqrt{3}}""\\theta=\\pi+\\tan^{-1}(-\\dfrac{1}{\\sqrt{3}})=\\dfrac{5\\pi}{6}""z_1=2(\\cos\\dfrac{5\\pi}{6}+i\\sin\\dfrac{5\\pi}{6})"





"r^2=(1)^2+(-1)^2=2""r\\ge0=>r=\\sqrt{2}"

Quadrant IV


"\\tan \\theta=\\dfrac{-1}{\\sqrt{3}}=-\\dfrac{1}{1}=-1""\\theta=2\\pi+\\tan^{-1}(-1)=\\dfrac{7\\pi}{4}""z_2=\\sqrt{2}(\\cos\\dfrac{7\\pi}{4}+i\\sin\\dfrac{7\\pi}{4})"

"z_1\/z_2"



"\\theta_{z_1\/z_2}=\\dfrac{5\\pi}{6}-\\dfrac{7\\pi}{4}=-\\dfrac{11\\pi}{12}"

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