InterviewSolution
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InterviewSolution
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Express the complex number `(-sqrt(3)-i)` in polar form. |
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Answer» The given complex number is `z = (-sqrt(3)-i)`. Let its polar form be `z=r(cos theta + i sin theta)`. Now, `r=|z|=sqrt((-sqrt(3))^(2)+(-1)^(2))=sqrt(4)=2`. Let `alpha` be the acute angle, given by `tan alpha=|(Im(z))/(Re(z))|=|(-1)/(-sqrt(3))|=(1)/(sqrt(3)) rArr alpha = (pi)/(6)`. Clearly, the point representing the complex number `z = (-sqrt(3)-i)` is `P(-sqrt(3), -1)`, which lies in the third quadrant. `therefore" "arg(z) = theta = -(pi-alpha)=-(pi-(pi)/(6))=(-5pi)/(6)`. Thus, `r=|z|=2 and theta = (-5pi)/(6)`. Hence, the polar form of `z = (-sqrt(3)-i)` is given by `z=2{cos((-5pi)/(6))+i sin((-5pi)/(6))}, i.e., 2("cos"(5pi)/(6)-"i sin"(5pi)/(6))`. |
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