The polar expression of a complex number in its rectangular form 4(cos120° + i sin120°) = -2 + 2√3 i.
Let Z = a + ib be a complex number in the cartesian form represented by point P(a, b) in the complex plane then Z = r cos∅ + i r sin∅ =r is called the polar form of a complex number.
The given expression is
z = 4(cos120° + i sin120°)
= 4(cos(90 + 30)° + i sin(90 + 30)°)
by trigonometric properties
cos (90 + ∅) = -sin ∅
sin (90 + ∅) = -cos ∅
so, 4( -sin (30)° + i cos (30)°)
sin (30)° = 1/2, cos (30)° = √3/2
Substitute the values
= 4 ( -1/2 + i √3/2)
= -4(-1/2) + i 4(√3/2)
= -2 + 2√3 i
Therefore, the rectangular form is 4(cos120° + i sin120°) = -2 + 2√3 i.
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