InterviewSolution
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InterviewSolution
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Convert each of the following complex numbers into polar form: `{:((i),3,(ii),-5,(iii),i,(iv),-2i):}` |
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Answer» (i) The given complex number is z = 3 + 0i. Let its polar form be `z = r(cos theta + i sin theta)`. Now, `r = |z| = sqrt(3^(2)+0^(2))=sqrt(9)=3`. Clearly, z = 3 + 0i is represented by the point P(3, 0), which lies on the positive side of the x-axis. `therefore" "arg(z) = theta = 0`. Thus, `r = 3 and theta = 0`. Hence, the required polar form of `z = 3 + 0i "is" 3(cos 0 + i sin 0)`. (ii) The given complex number is z = -5 + 0i. Let its polar form be `z = r(cos theta + i sin theta)`. Now, `r=|z|=sqrt((-5)^(2)+0^(2))=sqrt(25)=5`. Clearly, z = -5 + 0i is represented by the point P(-5, 0), which lies on the negative side of the x-axis. `therefore" "arg(z) = pi rArr theta = pi` Thus, r = 5 and `theta = pi`. Hence, the required polar form of z = -5 + 0i is `5(cos pi + i sin pi)`. (iii) The given complex number is z = 0 + i. Let its polar form be `z = r(cos theta + i sin theta)`. Now, `r = |z| = sqrt(0^(2)+1^(2))=1`. Clearly, z = 0 + i is respresented by the point P(0, 1), which lies on the y-axis and above the x-axis. `therefore" "arg(z)=(pi)/(2) rArr theta = (pi)/(2)`. Thus, `r = 1 and theta = (pi)/(2)`. Hence, the requird polar form of z = 0 + i is. `1.("cos"(pi)/(2)+"i sin"(pi)/(2)), i.e., ("cos"(pi)/(2)+"i sin"(pi)/(2))`. (iv) The given complex number is z = 0 -2i. Let its polar form be `z = r(cos theta + i sin theta)`. Now, `r = |z| = sqrt(0^(2) + (-2)^(2))=sqrt(4) =2`. Clearly, z = (0-2i) is represented by the point P(0, -2), which lies on the y-axis and below the x-axis. `therefore" "arg(z)=(-pi)/(2) rArr theta = (-pi)/(2)`. Thus, r = 2 and `theta = (-pi)/(2)`. Hence, the required polar form of z = 0 -2i is `z=2{cos((-pi)/(2))+i sin((-pi)/(2))}, i.e., 2("cos"(pi)/(2)-"i sin"(pi)/(2))`. |
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