Trigonometric Form of a Complex Number

Complex Number $a + b i$	Modulus $r$	Argument $θ$	Trigonometric Form $r (cos θ + i sin θ)$
$1 + i$	$2$	$\frac{π}{4}$	$2 (cos \frac{π}{4} + i sin \frac{π}{4})$
$- 4$	$4$	$π$	$4 (cos π + i sin π)$
$3 + i$	$2$	$\frac{π}{6}$	$2 (cos \frac{π}{6} + i sin \frac{π}{6})$

Complex Number

a + b i

Modulus

r

Argument

θ

Trigonometric Form

r (cos θ + i sin θ)

1 + i

2

\frac{π}{4}

2 (cos \frac{π}{4} + i sin \frac{π}{4})

- 4

4

π

4 (cos π + i sin π)

3 + i

2

\frac{π}{6}

2 (cos \frac{π}{6} + i sin \frac{π}{6})

Consider the nonzero complex number $z = a + b i$ on a complex plane.

Let $θ$ be the angle measured counterclockwise from the positive real axis to the line segment connecting the origin and the point, and let $r$ be the length of this segment.

The measure of angle

θ

is the arctangent of the ratio of the

y -

coordinate to the

x -

coordinate of the point.

θ = arctan (\frac{b}{a})

Since

r

is the distance of the complex number

z

from

(0, 0),

it can be calculated using the Distance Formula.

r = (a - 0)^{2} + (b - 0)^{2} = a^{2} + b^{2}

Now, the coordinates

a

and

b

of the point can be defined in terms of the polar coordinates

r

and

θ

as follows.

a b = r cos θ = r sin θ

These expressions are substituted for

a

and

b

into the standard form of a complex number.

z z = a + b i ⇓ = (r cos θ) + i (r sin θ)

By factoring out

r

from the expression, the trigonometric form of the number is obtained.

$z = r (cos θ + i sin θ)$

Principally, the angle $θ$ and modulus $r$ act like a compass and ruler, working together with sine and cosine functions to map each complex number's unique position on the plane.

Trigonometric Form of a Complex Number

Why

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