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What is the product (4√3-4i)(√2+√2i) in polar form? In what quadrant of the complex plane does the product lie?

 May 1, 2020
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To change  x + yi  into polar form, use these two formula:  r  =  sqrt( x2 + y2 )   and  theta  =  tan-1( y/x ).

Polar form is:  r·cis( theta)

 

4sqrt(3) - 4i  :  r  =  sqrt( [4sqrt(3)]2 + [-4]2 )  =  sqrt( 48 + 16 )  =  sqrt( 64 )  =  8

                       theta  =  tan-1( -4 / (4sqrt(3) )  =  tan-1( 1 / sqrt(3) )  =  tan-1( sqrt(3) / 3 )  =  -pi/6  or  11pi/6

--->   8·cis( -pi / 6)

 

sqrt(2) + sqrt(2)i  :  r  =  sqrt( [sqrt(2)2 ] + [sqrt(2)2​ ] )  =  sqrt( 2 + 2 )  =  sqrt( 4 )  =  2

                              theta  =  tan-1( sqrt(2) / sqrt(2) )  =  tan-1( 1 )  =  pi/4

--->   2·cis( pi/4)

 

The product of  a·cis(A) x b·cis(B)  =  a·b·is(A + B)

 

   8cis( -pi / 6) x 2cis( pi/4)  =  16cis( pi/12 )

 

This will be in the first quadrant.

 May 1, 2020

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