Find the two square roots of 36i.

Martin Hart 2022-10-22 Answered
Find the two square roots of 36i.
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Answers (1)

getrdone07tl
Answered 2022-10-23 Author has 23 answers
z = 36 i = 0 + 36 i
Now,
r = 0 2 + ( 36 ) 2 = 36
&
θ = tan 1 ( x y ) = tan 1 ( 36 0 ) = tan 1 ( ) = π 2
Now the polar form is,
z = r ( cos θ + i sin θ ) z = 36 ( cos π 2 + i sin π 2 )
Now square roots of z is,
z = [ 36 ( cos π 2 + i sin π 2 ) ] 1 2 = 6 ± ( cos π 2 + i sin π 2 ) 1 2 = 6 ± ( cos ( 1 2 π 2 ) + i sin ( 1 2 π 2 ) )       [ ( cos θ + i sin θ ) n = cos n θ + i sin n θ ] = 6 ± ( cos ( π 4 ) + i sin ( π 4 ) ) = 6 ± ( 2 2 + i 2 2 ) = ± ( 3 2 + 3 i 2 )
Hence the two square roots are,
3 2 3 i 2    &    3 2 + 3 i 2
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