What is the product (4√3-4i)(√2+√2i) in polar form? In what quadrant of the complex plane does the product lie?
+23246 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.
