works well for addition and subtraction, but becomes unwieldy when computing powers and roots. Trigonometric form offers an alternative description: instead of specifying horizontal and vertical components, we specify how far a number lies from the origin and at what angle. This polar perspective transforms multiplication into rotation and scaling, making operations that are tedious in algebraic form almost trivial.

Key Terms

Trigonometric Form— representation as

r(\cos\theta + i\sin\theta)

Modulus— the radius

r = |z|

Argument— the angle

\theta

determining direction

Algebraic Form— the rectangular counterpart

a + bi

Exponential Form— the compact equivalent

re^{i\theta}

See All Complex Numbers Definitions →

Why a New Representation?

Computing

(1 + i)^{10}

in algebraic form requires expanding a binomial to the tenth power — dozens of terms, careful bookkeeping, and ample opportunity for error. The trigonometric form reduces this calculation to a few seconds of mental arithmetic.

The key insight comes from the geometric representation of complex numbers. Every complex number corresponds to a point in the plane, and every point can be described two ways: by its Cartesian coordinates

(a, b)

or by its polar coordinates

(r, \theta)

. The algebraic form uses Cartesian coordinates. The trigonometric form uses polar coordinates.

Two parameters define a complex number in this representation:

Modulus ( $r$ ): The distance from the origin to the point, identical to the absolute value $|z|$ .

Argument ( $\theta$ ): The angle formed between the positive real axis and the line segment connecting the origin to the point.

This shift from rectangular to polar thinking unlocks powerful computational shortcuts. Multiplication becomes rotation combined with scaling. Division becomes counter-rotation combined with shrinking. Powers and roots reduce to simple arithmetic on the modulus and argument.

Modulus (Radius)

The modulus of a complex number

z = a + bi

measures its distance from the origin in the complex plane. This quantity is computed using the Pythagorean theorem:

r = |z| = \sqrt{a^2 + b^2}

The modulus is always a non-negative real number. It equals zero only when both

a = 0

and

b = 0

, meaning

z = 0

is the sole complex number with zero modulus.

Examples

For

z = 3 + 4i

r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

For

z = -2 + 2i

r = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}

For

z = -5i

r = \sqrt{0^2 + (-5)^2} = \sqrt{25} = 5

The modulus represents the "size" of a complex number without regard to its direction. Two numbers with the same modulus lie on a circle of radius

r

centered at the origin.

Argument (Angle)

The argument of a complex number

z

is the angle

\theta

measured from the positive real axis to the line segment connecting the origin to

z

. By convention, angles measured counterclockwise are positive, and angles measured clockwise are negative.

The notation

\arg(z) = \theta

denotes the argument of

z

— one of the standard complex number symbols. Arguments are typically expressed in radians, though degrees appear in elementary contexts.

Finding the Argument

For

z = a + bi

with

a \neq 0

, the tangent of the argument satisfies:

Argument

\tan\theta = \frac{b}{a}

Learn more about this formula: Argument →

Browse all complex numbers formulas →

However, the arctangent function alone does not determine

\theta

uniquely. The equation

\tan\theta = \frac{b}{a}

has solutions in two quadrants, so you must identify which quadrant contains

z

and select the appropriate angle.

The Argument Is Not Unique

Adding any integer multiple of

2\pi

to an argument produces another valid argument for the same complex number. If

\theta

is an argument of

z

, then so is

\theta + 2\pi k

for any integer

k

. The number

z = 1 + i

has argument

\frac{\pi}{4}

, but also

\frac{\pi}{4} + 2\pi = \frac{9\pi}{4}

, and

\frac{\pi}{4} - 2\pi = -\frac{7\pi}{4}

, and infinitely many others.

The Principal Argument

To avoid ambiguity, mathematicians define a unique representative called the principal argument. The principal argument of

z

, denoted

\text{Arg}(z)

with a capital A, is the unique argument lying in a specified standard interval.

Two conventions exist:

Convention 1: $\text{Arg}(z) \in (-\pi, \pi]$

Convention 2: $\text{Arg}(z) \in [0, 2\pi)$

The first convention is more common in advanced mathematics. Under this convention, arguments range from just above

-\pi

(pointing almost directly left, slightly below the axis) through

\pi

(pointing directly left).

Examples Using $(-\pi, \pi]$

For

z = 1

(positive real):

\text{Arg}(z) = 0

For

z = i

(positive imaginary):

\text{Arg}(z) = \frac{\pi}{2}

For

z = -1

(negative real):

\text{Arg}(z) = \pi

For

z = -i

(negative imaginary):

\text{Arg}(z) = -\frac{\pi}{2}

For

z = 1 + i

(first quadrant):

\text{Arg}(z) = \frac{\pi}{4}

For

z = -1 - i

(third quadrant):

\text{Arg}(z) = -\frac{3\pi}{4}

The principal argument provides a canonical choice when a unique angle is required.

Quadrant Considerations

The formula

\theta = \arctan(b/a)

returns values in

(-\frac{\pi}{2}, \frac{\pi}{2})

, which covers only the right half of the plane. For complex numbers in the left half-plane, adjustments are necessary.

Quadrant I: $a > 0$ , $b > 0$

The arctangent gives the correct angle directly:

\theta = \arctan\left(\frac{b}{a}\right)

Quadrant II: $a < 0$ , $b > 0$

Add

\pi

to the arctangent result:

\theta = \arctan\left(\frac{b}{a}\right) + \pi

Quadrant III: $a < 0$ , $b < 0$

Subtract

\pi

from the arctangent result (for principal argument in

(-\pi, \pi]

\theta = \arctan\left(\frac{b}{a}\right) - \pi

Quadrant IV: $a > 0$ , $b < 0$

The arctangent gives the correct angle directly (it will be negative):

\theta = \arctan\left(\frac{b}{a}\right)

Alternative: The atan2 Function

Many calculators and programming languages provide

\text{atan2}(b, a)

, which automatically handles quadrant adjustment and returns the principal argument directly.

Quadrant	Signs of (a, b)	Argument formula (principal arg in (−π, π])
I	a > 0, b > 0	θ = arctan(b ⁄ a)
II	a < 0, b > 0	θ = arctan(b ⁄ a) + π
III	a < 0, b < 0	θ = arctan(b ⁄ a) − π
IV	a > 0, b < 0	θ = arctan(b ⁄ a) (result is negative)

Special Angles on the Axes

When a complex number lies on one of the coordinate axes, the argument takes a standard value that requires no calculation.

Positive Real Axis

For

z = a

where

a > 0

\theta = 0

The number points directly to the right.

Positive Imaginary Axis

For

z = bi

where

b > 0

\theta = \frac{\pi}{2} \quad (90°)

The number points directly upward.

Negative Real Axis

For

z = a

where

a < 0

\theta = \pi \quad (180°)

The number points directly to the left.

Negative Imaginary Axis

For

z = bi

where

b < 0

\theta = -\frac{\pi}{2} \quad (-90°) \quad \text{or equivalently} \quad \frac{3\pi}{2} \quad (270°)

The number points directly downward. Which value you use depends on your chosen interval for the principal argument.

The Origin

For

z = 0

, the argument is undefined. A point at the origin has no direction from itself.

Position	Example	Argument (radians)	Argument (degrees)
Positive real axis	z = a, a > 0	0	0°
Positive imaginary axis	z = bi, b > 0	π ⁄ 2	90°
Negative real axis	z = a, a < 0	π	180°
Negative imaginary axis	z = bi, b < 0	−π ⁄ 2 (or 3π ⁄ 2)	−90° (or 270°)

The Trigonometric Form

With modulus

r

and argument

\theta

defined, the trigonometric form expresses a complex number as:

Trigonometric Form

z = r(\cos\theta + i\sin\theta)

Learn more about this formula: Trigonometric Form →

Browse all complex numbers formulas →

This formula encodes the relationship between polar and Cartesian coordinates. The real part is

a = r\cos\theta

and the imaginary part is

b = r\sin\theta

The cis Notation

Because the combination

\cos\theta + i\sin\theta

appears constantly, an abbreviated notation exists:

\text{cis}\,\theta = \cos\theta + i\sin\theta

Using this shorthand, the trigonometric form becomes:

z = r\,\text{cis}\,\theta

The notation "cis" stands for "cosine + i sine." It appears frequently in precalculus and engineering contexts.

Connection to Exponential Form

The trigonometric form leads directly to the exponential form through Euler's formula:

e^{i\theta} = \cos\theta + i\sin\theta

This identity shows that

\text{cis}\,\theta = e^{i\theta}

, so the trigonometric and exponential forms are two notations for the same representation.

Trigonometric Form Notation

Notation

Trigonometric Form Notation

The angle marks live here: the multivalued operator, the capital that picks one value, the polar letter pair, and the abbreviation that packs cosine-plus-i-sine into three letters. The bar of

|z|

and the exponential

e^{i\theta}

each have their own page.

z

a + bi

and

\mathbb{C}

come from complex numbers basics,

i

from the imaginary unit; the modulus bars

|z|

are owned by the absolute value page,

e^{i\theta}

by the exponential form.

\arg(z)

arg of z — the argument of z

The angle operator:

\arg(z) = \theta

, the counterclockwise angle from the positive real axis to

z

. Genuinely multivalued — if

\theta

works, so does

\theta + 2\pi k

for every integer

k

; the non-uniqueness in Argument (Angle) above is the point, not a defect.

CasesRadians by default, degrees in elementary work — the degree and radian conventions carry over unchanged. Print sets

\arg

upright, like

\sin

and

\log

Also written

\varphi

\phi

— the phase, standard in physics and engineering, especially for phasors and waves.

Do not confuseA single value. Treating

\arg(z)

as one number produces the classic "paradoxes" around

\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)

— an identity that holds only up to multiples of

2\pi

\operatorname{Arg}(z)

capital arg of z — the principal argument

The capital letter buys uniqueness:

\operatorname{Arg}(z)

is the single representative of

\arg(z)

drawn from one agreed interval — a choice, not a discovery.

CasesTwo rival conventions:

(-\pi, \pi]

in advanced mathematics,

[0, 2\pi)

in elementary and computational settings — The Principal Argument above works the first. An answer of

-\frac{3\pi}{4}

under one is

\frac{5\pi}{4}

under the other; state the interval before comparing results.

Also written

\operatorname{atan2}(b, a)

— the programming spelling, returning the principal value directly; note the argument order,

b

before

a

, opposite the fraction in

\tan\theta = b/a

Do not confuse

\arg(z)

itself. Lowercase is the infinite family, uppercase the chosen one — mixing them silently is how off-by-

2\pi

errors enter multi-step calculations.

r

\theta

r and theta — the polar pair

The letter pair of polar description:

r = |z|

, the modulus, and

\theta = \arg(z)

. Together they rebuild the number —

z = r(\cos\theta + i\sin\theta)

, with

a = r\cos\theta

and

b = r\sin\theta

Cases

r

is non-negative by definition — a "negative

r

" is not alternative notation but a different number, its

\theta

shifted by

\pi

. Subscripts track multiple numbers:

r_1 r_2

and

\theta_1 + \theta_2

in Multiplication and Division in Trigonometric Form below.

Also written

\rho

(rho) for the modulus in some European and physics texts, freeing

r

for other radii; the pair

(r, \theta)

deliberately mirrors polar coordinates.

Do not confuseA degree–radian mix. The pair means nothing until the angle unit is fixed; converting

r

correctly and

\theta

in the wrong mode is the classic calculator error.

\operatorname{cis}\theta

sis theta — cosine plus i sine

A pure abbreviation:

\operatorname{cis}\theta = \cos\theta + i\sin\theta

, so

z = r\operatorname{cis}\theta

. It adds no mathematics — it compresses the bracket that The Trigonometric Form above spells out.

CasesAt home in precalculus and engineering texts; research mathematics skips it and writes

e^{i\theta}

via Euler's formula instead — same object, one notation generation apart.

Also written

r\angle\theta

— electrical engineering's phasor angle notation, standard on circuit diagrams and calculator displays.

Do not confuseA product.

\operatorname{cis}

is one operator, not

c \cdot i \cdot s

, and

\operatorname{cis}(n\theta) \neq n\,\operatorname{cis}(\theta)

— the argument lives inside the operator, which is exactly what De Moivre's theorem exploits.

All complex number symbols →↑ Complex Numbers

Converting Algebraic to Trigonometric

To convert

z = a + bi

into trigonometric form

z = r\,\text{cis}\,\theta

:

Step 1: Calculate the modulus.

r = \sqrt{a^2 + b^2}

Step 2: Determine the argument using

\tan\theta = \frac{b}{a}

with appropriate quadrant adjustment.

Step 3: Write the result as

z = r\,\text{cis}\,\theta

z = r(\cos\theta + i\sin\theta)

Example 1: Convert $z = 1 + \sqrt{3}i$

Step 1:

r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2

Step 2:

\tan\theta = \frac{\sqrt{3}}{1} = \sqrt{3}

. Since

a > 0

and

b > 0

(first quadrant),

\theta = \arctan(\sqrt{3}) = \frac{\pi}{3}

.

Step 3:

z = 2\,\text{cis}\,\frac{\pi}{3}

Example 2: Convert $z = -3 - 3i$

Step 1:

r = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}

Step 2:

\tan\theta = \frac{-3}{-3} = 1

. Since

a < 0

and

b < 0

(third quadrant),

\theta = \arctan(1) - \pi = \frac{\pi}{4} - \pi = -\frac{3\pi}{4}

.

Step 3:

z = 3\sqrt{2}\,\text{cis}\left(-\frac{3\pi}{4}\right)

Converting Trigonometric to Algebraic

To convert

z = r\,\text{cis}\,\theta

into algebraic form

z = a + bi

:

Step 1: Calculate the real part.

a = r\cos\theta

Step 2: Calculate the imaginary part.

b = r\sin\theta

Step 3: Write the result as

z = a + bi

Example 1: Convert $z = 4\,\text{cis}\,60°$

Step 1:

a = 4\cos 60° = 4 \cdot \frac{1}{2} = 2

Step 2:

b = 4\sin 60° = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}

Step 3:

z = 2 + 2\sqrt{3}i

Example 2: Convert $z = 5\,\text{cis}\,\frac{3\pi}{2}$

Step 1:

a = 5\cos\frac{3\pi}{2} = 5 \cdot 0 = 0

Step 2:

b = 5\sin\frac{3\pi}{2} = 5 \cdot (-1) = -5

Step 3:

z = 0 + (-5)i = -5i

Example 3: Convert $z = 2\,\text{cis}\,\frac{5\pi}{6}$

Step 1:

a = 2\cos\frac{5\pi}{6} = 2 \cdot \left(-\frac{\sqrt{3}}{2}\right) = -\sqrt{3}

Step 2:

b = 2\sin\frac{5\pi}{6} = 2 \cdot \frac{1}{2} = 1

Step 3:

z = -\sqrt{3} + i

Multiplication and Division in Trigonometric Form

The trigonometric form reveals that multiplication of complex numbers combines rotation with scaling, while division reverses both operations.

Multiplication Rule

For

z_1 = r_1\,\text{cis}\,\theta_1

and

z_2 = r_2\,\text{cis}\,\theta_2

Multiplication in Polar Form

z_1 \cdot z_2 = r_1 r_2\,\text{cis}(\theta_1 + \theta_2)

Learn more about this formula: Multiplication in Polar Form →

Browse all complex numbers formulas →

Multiply the moduli and add the arguments. Geometrically, multiplying by

z_2

scales distances by

r_2

and rotates by angle

\theta_2

Division Rule

For

z_1 = r_1\,\text{cis}\,\theta_1

and

z_2 = r_2\,\text{cis}\,\theta_2

with

z_2 \neq 0

Division in Polar Form

\frac{z_1}{z_2} = \frac{r_1}{r_2}\,\text{cis}(\theta_1 - \theta_2)

Learn more about this formula: Division in Polar Form →

Browse all complex numbers formulas →

Divide the moduli and subtract the arguments.

Example: Multiplication

Let

z_1 = 2\,\text{cis}\,30°

and

z_2 = 3\,\text{cis}\,45°

z_1 \cdot z_2 = (2)(3)\,\text{cis}(30° + 45°) = 6\,\text{cis}\,75°

Example: Division

Let

z_1 = 10\,\text{cis}\,\frac{2\pi}{3}

and

z_2 = 2\,\text{cis}\,\frac{\pi}{6}

\frac{z_1}{z_2} = \frac{10}{2}\,\text{cis}\left(\frac{2\pi}{3} - \frac{\pi}{6}\right) = 5\,\text{cis}\,\frac{\pi}{2}

These formulas extend naturally to De Moivre's theorem, which handles powers and roots with equal elegance.

Trigonometric Form at a Glance

The quantities and operations covered in the sections above collect into a single reference card. Each row gives a quantity or operation and its formula in trigonometric form — useful as a study summary and as a quick lookup when computing modulus, argument, conversions, products, or quotients.

Quantity / operation	Formula
Modulus r = \|z\|	r = √(a² + b²)
Argument θ	tan θ = b ⁄ a, with quadrant adjustment (or atan2(b, a))
Trigonometric form	z = r(cos θ + i sin θ) = r cis θ
Algebraic → trigonometric	r = √(a² + b²); θ from quadrant of (a, b)
Trigonometric → algebraic	a = r cos θ, b = r sin θ
Multiplication z₁ · z₂	r₁r₂ cis(θ₁ + θ₂) — multiply moduli, add arguments
Division z₁ ⁄ z₂	(r₁ ⁄ r₂) cis(θ₁ − θ₂) — divide moduli, subtract arguments
Exponential equivalent	r cis θ = re^iθ (Euler's formula)

Trigonometric Form FAQ

What is the argument of a complex number?

The angle from the positive real axis to the line joining the origin to the point, measured counterclockwise. Together with the modulus it locates the number completely, which is what polar coordinates do for any point in the plane. It is undefined at the origin, where no direction exists.Read more →

Why is the argument not unique?

Because adding a full turn returns to the same direction, so every number has infinitely many valid arguments differing by multiples of two pi. To make it single-valued, one is chosen as the principal argument, conventionally the one lying between negative pi and pi. Formulas often need a full turn added or removed to land back inside that range.Read more →

How do you find the argument in each quadrant?

The inverse tangent of the ratio gives the right answer only in the first and fourth quadrants, since it cannot tell opposite directions apart. In the second and third, pi must be added or subtracted to correct it. Sketching the point before trusting the calculator is the reliable way to avoid the error.Read more →

What is cis notation?

A shorthand for cosine plus i times sine of the same angle, so cis theta compresses the whole trigonometric form into one symbol. It saves writing but is falling out of use in favour of the exponential form, which conveys the same content and behaves algebraically like an ordinary power. Convert between forms on the polar-rectangular tool.Read more →

How do you convert from algebraic to trigonometric form?

Compute the modulus from the two components using the Pythagorean relationship, then find the angle from their ratio and correct it for the quadrant. Substituting both into the standard form completes the conversion. Going the other way is easier: multiply the modulus by the cosine and sine to recover the components.Read more →

Trigonometric Form

Describing Complex Numbers by Distance and Direction

Key Terms

Why a New Representation?

Modulus (Radius)

Examples

Argument (Angle)

Finding the Argument

The Argument Is Not Unique

The Principal Argument

Examples Using (−π,π](-\pi, \pi](−π,π]

Quadrant Considerations

Quadrant I: a>0a > 0a>0, b>0b > 0b>0

Quadrant II: a<0a < 0a<0, b>0b > 0b>0

Quadrant III: a<0a < 0a<0, b<0b < 0b<0

Quadrant IV: a>0a > 0a>0, b<0b < 0b<0

Alternative: The atan2 Function

Special Angles on the Axes

Positive Real Axis

Positive Imaginary Axis

Negative Real Axis

Negative Imaginary Axis

The Origin

The Trigonometric Form

The cis Notation

Connection to Exponential Form

Trigonometric Form Notation

Trigonometric Form Notation

Converting Algebraic to Trigonometric

Example 1: Convert z=1+3iz = 1 + \sqrt{3}iz=1+3​i

Example 2: Convert z=−3−3iz = -3 - 3iz=−3−3i

Converting Trigonometric to Algebraic

Example 1: Convert z=4 cis 60°z = 4\,\text{cis}\,60°z=4cis60°

Example 2: Convert z=5 cis 3π2z = 5\,\text{cis}\,\frac{3\pi}{2}z=5cis23π​

Example 3: Convert z=2 cis 5π6z = 2\,\text{cis}\,\frac{5\pi}{6}z=2cis65π​

Multiplication and Division in Trigonometric Form

Multiplication Rule

Division Rule

Example: Multiplication

Example: Division

Trigonometric Form at a Glance

Trigonometric Form FAQ

What is the argument of a complex number?

Why is the argument not unique?

How do you find the argument in each quadrant?

What is cis notation?

How do you convert from algebraic to trigonometric form?

Examples Using $(-\pi, \pi]$

Quadrant I: $a > 0$ , $b > 0$

Quadrant II: $a < 0$ , $b > 0$

Quadrant III: $a < 0$ , $b < 0$

Quadrant IV: $a > 0$ , $b < 0$

Example 1: Convert $z = 1 + \sqrt{3}i$

Example 2: Convert $z = -3 - 3i$

Example 1: Convert $z = 4\,\text{cis}\,60°$

Example 2: Convert $z = 5\,\text{cis}\,\frac{3\pi}{2}$

Example 3: Convert $z = 2\,\text{cis}\,\frac{5\pi}{6}$