Question

Convert the complex number z=1+3i/1-2i
into polar form
​

Answer 1

see the attachment

hope it helps you !!

Answer 2

The answer is √2 (cos 3π/4 +i sin 3π/4 )

Here, z= 1+3i/1−2i

= 1+3i/1−2i × 1+2i/1+2i

​= 1+2i+3i−6/1+4

= −5+5i/5

=1+i

Let rcosθ=−1 and rsinθ =1

on squaring and adding. we obtain

r 2 (cos 2 θ+sin 2 θ) =1+1

⇒r 2 (cos 2 θ+sin 2 θ)=2

⇒r2 = 2 [cos 2 θ+sin 2 θ=1]

⇒r= √2 [Conventionally, r>0]

∴ √2cosθ=−1 and  √2sinθ=1

⇒cosθ= = -1/√2  and sinθ = 1/√

∴z=rcosθ+i rsinθ

= √2 cos 3π/4 +i √2 sin 3π/4

= √2 (cos 3π/4 +i sin 3π/4 )

This is the required polar form.

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