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