Trig Form of a Complex Number

[Jessica]

The absolute value of the complex number a + bi is defined as the distance between the origin (0,0) and the point (a, b)

|a + bi| =




Using the complex number a + bi, we can let θ be the angle from the positive real axis (counterclockwise) to the line segment connecting the origin to the point (a, b). Therefore, a = r cosθ and b = r sinθ where r =

TRIG FORM OF A COMPLEX NUMBER

z = r(cos θ + i sin θ)

*Abbreviated form: z = rcisθ



Product and Quotient of two complex numbers

Let z[sub]1[/sub] = r[sub]1[/sub](cos θ[sub]1[/sub] + i sin θ[sub]1[/sub]) and z[sub]2[/sub] = r[sub]2[/sub](cos θ[sub]2[/sub] + i sin θ[sub]2[/sub]) be complex numbers

z[sub]1[/sub]z[sub]2[/sub] = r[sub]1[/sub]r[sub]2[/sub][cos(θ[sub]1[/sub] + θ[sub]2[/sub]) + i sin(θ[sub]1[/sub] + θ[sub]2[/sub])]

= [cos(θ[sub]1[/sub] − θ[sub]2[/sub]) + i sin(θ[sub]1[/sub] − θ[sub]2[/sub])]

DeMoivre's Theorem - Powers of Complex Numbers

If z = r(cos θ + i sin θ) is a complex number and n is a positive integer, then