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What is the complex number  z=−3+3√3 i represented in polar form?

Enter your answer by filling in the boxes. Enter the angle measure as the smallest positive angle in simplified form and any radicals in simplified form.

z=___cis(___)

 May 14, 2020
 #1
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Using trig we can make this triangle:

 

 

 

z  =  -3  +  3√3 i

                                                              Substitute   6 cos( 2π/3 )   in for   -3   and   6 sin( 2π/3 )   in for   3√3

z   =   6 cos( 2π/3 )  +  6 sin( 2π/3 ) i

                                                              Move the  i  to the front of the term

z   =   6 cos( 2π/3 )  +  6 i sin( 2π/3 )

                                                              Factor  6  out of both terms

z   =   6 [ cos( 2π/3 )  +  i sin( 2π/3 ) ]

                                                              cis( x )  =  cos( x ) + i sin( x )    so    cos( 2π/3 )  +  i sin( 2π/3 )  =  cis( 2π/3 )

z   =   6 cis( 2π/3 )

 May 14, 2020

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