Question

Express the following complex numbers in the polar form
$(\frac{2+i}{3-i} )^{2}$

Answer 1

Answer:

The given complex number is z=(−2+2i3–√).

Let its polar form be z=r(cosθ+isinθ).

Now, r=|z|=(−2)2+(23–√)2−−−−−−−−−−−−−−√=4+12−−−−−√=16−−√=4.

Let α be the acute angle, given by

tanα=∣∣∣Im(z)Re(z)∣∣∣=∣∣∣23–√−2∣∣∣=3–√⇒α=π3.

Clearly, the point representing z=(−2+2i3–√)is P(−2,23–√), which lies in the second quadrant.

∴ arg(z)=θ=(π−α)=(π−π3)=2π3.

Thus, r=|z|=4andθ=2π3.

Hence, the required polar form is =4(cos2π3+i sin2π3)

Answer 2

Answer:

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