How do you write the following in trigonometric form and perform the operation given 3+4i1−√3i?

1 Answer
Salvatore I.
Nov 13, 2016

z=34−2√3+i(342√3+1)=52(cosθ+isinθ)
with θ=arctan(25162√3+3−3916)

Explanation:

let's get rid of complex denominator...
z=3+4i1−i2√31+i2√31+i2√3=3+i32√3+i4−42√34
z=34−2√3+i(342√3+1) that in trigonometric form is z=|z|(cosθ+isinθ)
|z|=2√(34−2√3)2+(342√3+1)2
|z|=2√916+3−322√3+2716+1+322√3=
|z|=2√36+6416=104=52
the angle is θ=arctan(342√3+134−2√3)
θ=arctan(342√3+134−2√3⋅34+2√334+2√3)=

θ=arctan(9162√3+94+34+2√3916−3)
θ=arctan(25162√3+3−3916)

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