Trigonometric Form of Complex Numbers

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:

Modulus (Radius)

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

RULE

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

:

Modulus (Radius)

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

For

z = -2 + 2i

:

Modulus (Radius)

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

For

z = -5i

:

Modulus (Radius)

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

Pitfall: Squaring bi instead of b

For

z = 3 + 4i

, putting

4i

into the formula gives

\sqrt{3^2 + (4i)^2} = \sqrt{9 - 16} = \sqrt{-7}

, not a real number. The formula takes

b = 4

, the coefficient of

i

:

r = \sqrt{9 + 16} = 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.

z = 3 + 2i: r = √13 ≈ 3.61

The modulus is the length of the arrow from the origin, the hypotenuse of the right triangle with legs a = 3 and b = 2, so r = √(9 + 4). It is the first of the two polar coordinates. Read r and θ for any point on the polar and rectangular form converter.

Every complex number except zero has a positive modulus, and zero alone has no direction.

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

RULE

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.

z = −4 + 3i: the angle θ measured from the positive real axis

The argument is the counterclockwise angle from the positive real axis to the arrow; for this point in the second quadrant it lies between 90° and 180°, about 143°. Any angle differing from it by a full turn names the same point. Watch θ change as the point moves on the polar and rectangular form converter.

The principal argument, treated next, picks one representative from this infinite family.

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

Fact: Every angle choice jumps somewhere

No rule that picks one argument for every nonzero

z

can avoid a jump. Follow a point once around the origin and its angle grows by

2\pi

, so a single-valued choice must drop back by

2\pi

somewhere; a convention only decides where.

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).

The two ends of the interval meet on the negative real axis. A point just above that axis has principal argument close to

\pi

, while a point just below it has principal argument close to

-\pi

, even though the two points almost touch. Under the second convention the same break sits on the positive real axis, between angles near

0

and angles near

2\pi

.

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.

A negative principal argument

z = 5 − 5i lies in the fourth quadrant. The angle arc runs clockwise from the positive real axis, a quarter of a half turn: under the (−π, π] convention Arg(z) = −π/4, not 7π/4. Both angles name the same point; the principal argument is the one inside the standard interval. Drag z into other quadrants on the polar and rectangular converter.

Under the [0, 2π) convention the same z has Arg(z) = 7π/4: the point is fixed, only the chosen representative changes.

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:

Quadrant Considerations

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

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

Add

\pi

to the arctangent result:

Quadrant Considerations

\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]

):

Quadrant Considerations

\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):

Quadrant Considerations

\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)

z = −3 − 4i: arctan(b/a) alone gives the wrong angle

Both parts are negative, so arctan(−4/−3) = arctan(4/3) ≈ 53° points into the first quadrant, while the arrow plainly lies in the third; adding 180° gives the correct argument of about 233°. The quadrant of the point, not the formula, decides. Move the point between quadrants II and III on the polar and rectangular form converter.

The rule is simple: compute the reference angle, then place it in the quadrant the signs of a and b dictate.

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

:

Special Angles on the Axes

\theta = 0

The number points directly to the right.

Positive Imaginary Axis

For

z = bi

where

b > 0

:

Special Angles on the Axes

\theta = \frac{\pi}{2} \quad (90°)

The number points directly upward.

Negative Real Axis

For

z = a

where

a < 0

:

Special Angles on the Axes

\theta = \pi \quad (180°)

The number points directly to the left.

Negative Imaginary Axis

For

z = bi

where

b < 0

:

Special Angles on the Axes

\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

The algebraic form finds a point by going across and then up.

1 + i

means: one unit along the real axis, then one unit up. The two numbers

a = 1

and

b = 1

are directions for reaching the point at

(1, 1)

.

The same point can be found another way: by how far away it is and in which direction.

Stand at the origin, turn to face the point, and walk to it. That takes two numbers as well: the distance, which is the modulus

r

, and the angle turned from the positive real axis, which is the argument

\theta

. For

1 + i

the distance is

\sqrt{1^2 + 1^2} = \sqrt{2}

and the angle is

\frac{\pi}{4}

, halfway between the two axes.

Both descriptions of 1 + i lead to the same place.

Turn the distance and angle back into across and up with cosine and sine, and the first description comes out exactly.

description of 1 + i	the two numbers	across	up
across and up	a = 1, b = 1	1	1
distance and angle	r = √2, θ = π/4	√2 · cos(π/4) = 1	√2 · sin(π/4) = 1

They had to agree: across, up and the distance form one right triangle.

The arrow from the origin to any point is the long side of a right triangle whose other two sides are the across part

a

and the up part

b

. With the angle

\theta

at the origin, cosine and sine say exactly

a = r\cos\theta

and

b = r\sin\theta

— for every point, not just

1 + i

.

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 definition

Writing

a + bi

through its modulus and argument,

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

, gives the trigonometric form of the complex number.

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 →

So

1 + i = \sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right)

.

It is the same number in a different notation, and the notation has shorter versions.

The cis Notation

Because the combination

\cos\theta + i\sin\theta

appears constantly, an abbreviated notation exists:

The Trigonometric Form

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

Using this shorthand, the trigonometric form becomes:

The Trigonometric Form

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:

The Trigonometric Form

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.

Which description to reach for depends on the job — and the trigonometric one has a strict shape.

Adding is easiest across and up; multiplying and taking powers are easiest by distance and angle, which is how

(1 + i)^{10}

becomes quick. The section on multiplication and division in trigonometric form shows how.

task	easier in	why
add or subtract	algebraic form	add the parts
multiply or divide	trigonometric form	multiply the moduli, add the angles
powers such as (1 + i)¹⁰	trigonometric form	raise r, multiply θ

The form only reads correctly as written:

r

positive, the same angle in both places, and a plus sign between them.

Pitfall: Reading the argument through a minus sign

Reading

2\left(\cos\frac{\pi}{3} - i\sin\frac{\pi}{3}\right)

as argument

\frac{\pi}{3}

ignores the minus sign; the form needs a plus. Since cosine is even and sine is odd, the number is

2\left(\cos\left(-\frac{\pi}{3}\right) + i\sin\left(-\frac{\pi}{3}\right)\right)

, with argument

-\frac{\pi}{3}

.

The rest of the page covers the notation, converting between the algebraic and trigonometric forms in both directions, and multiplying and dividing in trigonometric form.

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

or

\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

, the same order as 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

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Notation · Cheat sheet

Complex Number Notation at a Glance

The marks from the notation cards across the complex-number pages: how each is read and what it means. Every group links to the full card on its page.

Basics

z, \; w

z, w: complex numbers

The usual letters for complex numbers, as

x

and

y

are for reals.

a + bi

a plus b i

Real part

a

, imaginary part

b

; both are real numbers.

\mathbb{C}

C, the complex numbers

The set of all

a + bi

; it contains

\mathbb{R}

.

Imaginary numbers

i

i, the imaginary unit

The number with

i^2 = -1

.

\sqrt{-1}

the square root of minus one

Another name for

i

; root rules for reals fail with it.

bi

b i

A pure imaginary number: real part

0

.

Algebraic form

\operatorname{Re}(z), \; \operatorname{Im}(z)

real part, imaginary part

For

z = a + bi

:

a

and

b

.

\operatorname{Im}(z)

is

b

, not

bi

.

a + (-b)i

a plus negative b, i

Written

a - bi

; the imaginary part is

-b

.

z = x + yi

z equals x plus y i

Same form with plane letters: the point

(x, y)

.

Complex conjugate

\bar{z}

z bar, the conjugate

For

z = a + bi

,

\bar{z} = a - bi

: the mirror image across the real axis.

\overline{z_1 + z_2}

the conjugate of the sum

The bar covers the whole expression; it equals

\bar{z}_1 + \bar{z}_2

.

z^*

z star

The same conjugate, as written in physics and engineering.

Absolute value

|z|

the modulus of z

Distance from

0

:

\sqrt{a^2 + b^2}

.

|z|^2 = z\bar{z}

mod z squared

Equals

a^2 + b^2

, with no root to take.

|z - c| = R

mod z minus c equals R

The circle of radius

R

around

c

.

Additive inverse

-z

negative z

Both parts change sign:

-a - bi

.

-(a + bi)

minus, quantity a plus b i

The brackets carry the minus to both parts.

z - w

z minus w

Adding the inverse:

z + (-w)

.

Multiplicative inverse

z^{-1}, \; \frac{1}{z}

z inverse; one over z

The number with

z \cdot z^{-1} = 1

; it equals

\frac{\bar{z}}{|z|^2}

.

(3 + 2i)^{-1}

the inverse of three plus two i

Found by multiplying top and bottom by the conjugate:

\frac{3 - 2i}{13}

.

Trigonometric formThis page

r, \; \theta

r and theta, the polar pair

Distance from

0

and angle from the positive real axis.

\arg(z)

the argument of z

Any angle of

z

; there are infinitely many,

2\pi

apart.

\operatorname{Arg}(z)

the principal argument

One chosen angle, usually in

(-\pi, \pi]

; some texts use

[0, 2\pi)

.

\operatorname{cis}\theta

sis theta

Short for

\cos\theta + i\sin\theta

.

Exponential form

e^{i\theta}

e to the i theta

Equals

\operatorname{cis}\theta

(Euler's formula): a point on the unit circle.

re^{i\theta}

r e to the i theta

The whole number

z

: length

r

, angle

\theta

.

e^{i(\theta + 2\pi k)}

theta plus two pi k

The same number for every integer

k

.

De Moivre's theorem

z_k

z sub k, the k-th root

The roots of

z^n = w

, numbered

k = 0, \ldots, n - 1

.

\omega

omega

A primitive root of unity: its powers give all

n

roots of

1

.

R^{1/n}

R to the one over n

The real root of the length; the complex root

w^{1/n}

has

n

values.

Same marks, different job

|z|

The modulus here; for a real number, the absolute value; for a matrix, the determinant.

\bar{z}

The conjugate here; in statistics a bar means a mean, in sets a complement.

z^{-1}

vs

z^*

The reciprocal, against the conjugate: they agree only on the unit circle.

w^{1/n}

n

values for a complex

w

; a single real root for a positive real number.

Complex number symbols table Complex numbers

Complex Number Notation at a Glance·/complex-numbers/basicsLearn Math Class

Converting Algebraic to Trigonometric

To convert

z = a + bi

into trigonometric form

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

:

Step 1: Calculate the modulus.

Converting Algebraic to Trigonometric

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

or

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)

Pitfall: Skipping the quadrant check

For

z = -1 + \sqrt{3}i

, taking

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

gives

2\,\text{cis}\left(-\frac{\pi}{3}\right) = 1 - \sqrt{3}i

, a different point. Here

a < 0

and

b > 0

, so

\theta = -\frac{\pi}{3} + \pi = \frac{2\pi}{3}

and

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

.

In both examples the modulus comes straight from the formula, and the real work is in Step 2. The ratio

\frac{b}{a}

fixes the tangent of the angle, while the signs of

a

and

b

fix the quadrant the angle lies in. Example 1 sits in the first quadrant, Example 2 in the third.

Converting Trigonometric to Algebraic

To convert

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

into algebraic form

z = a + bi

:

Step 1: Calculate the real part.

Converting Trigonometric to Algebraic

a = r\cos\theta

Step 2: Calculate the imaginary part.

Converting Trigonometric to Algebraic

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

Pitfall: Calculator in the wrong angle mode

Evaluating

4\cos 60°

with the calculator set to radians computes

4\cos(60 \text{ rad}) \approx -3.81

and

4\sin(60 \text{ rad}) \approx -1.22

. In degree mode the parts are

a = 2

and

b = 2\sqrt{3}

, as in Example 1.

The three examples use both angle units: the first angle is given in degrees, the other two in radians. The cosine and sine are evaluated at the angle exactly as written, and the modulus then scales both results. Exact values such as

\cos 60° = \frac{1}{2}

or

\sin\frac{3\pi}{2} = -1

keep the answer exact.

This direction needs no quadrant check. Cosine and sine already carry the right signs: in Example 3 the negative real part

-\sqrt{3}

comes out of

\cos\frac{5\pi}{6}

by itself and places the point in the second quadrant.

From angle to coordinates is plain evaluation; only the trip from coordinates back to an angle needs care with quadrants.

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

.

Fact: Two trig formulas in one product

The rule says the same product is

\cos(\alpha + \beta) + i\sin(\alpha + \beta)

, so its two parts are the angle-addition formulas of trigonometry. One complex multiplication carries both, and rebuilds them whenever they are forgotten.

For two numbers of modulus

1

, the rule can be set against ordinary expansion. Multiplying out

(\cos\alpha + i\sin\alpha)(\cos\beta + i\sin\beta)

term by term, with

i^2 = -1

, gives the real part

\cos\alpha\cos\beta - \sin\alpha\sin\beta

and the imaginary part

\sin\alpha\cos\beta + \cos\alpha\sin\beta

. Nothing about angles was used in this expansion, only the algebra of brackets.

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.

z₁z₂: arguments add, moduli multiply

The product sits at the angle θ₁ + θ₂ marked in the picture and at distance |z₁||z₂| from the origin: multiplying by a complex number rotates by its argument and scales by its modulus. Nothing in the algebraic expansion shows this; the picture does. Turn one factor and watch the product swing round on the complex multiplication tool.

This is the payoff of the trigonometric form: multiplication becomes a rotation and a scaling.

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)

Common Pitfalls

Four mistakes account for many of the wrong answers in work with the trigonometric form. Each is marked where it happens in the sections above; here each is worked through in full, with the wrong and the right working side by side.

Squaring bi instead of b

The modulus formula

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

uses the real number

b

, the coefficient of

i

. Squaring the whole imaginary term

bi

brings

i^2 = -1

into the formula and turns the sum under the root into a difference.

Wrong working: for

z = 3 + 4i

,

r = \sqrt{3^2 + (4i)^2} = \sqrt{9 - 16} = \sqrt{-7}

.

Right working:

b = 4

, so

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

.

Why it happens:

b

is read as the imaginary term

4i

rather than its coefficient

4

.

How to check: a modulus is a distance, so it is a non-negative real number. A negative number under the root shows the slip at once.

Reading the argument through a minus sign

The trigonometric form is

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

, with a plus sign and the same angle in both places. An expression with a minus sign between the terms looks like trigonometric form but is not, and reading

\theta

off it gives the wrong argument.

Wrong working:

2\left(\cos\frac{\pi}{3} - i\sin\frac{\pi}{3}\right)

has modulus

2

and argument

\frac{\pi}{3}

.

Right working: since

\cos(-\theta) = \cos\theta

and

\sin(-\theta) = -\sin\theta

, the number equals

2\left(\cos\left(-\frac{\pi}{3}\right) + i\sin\left(-\frac{\pi}{3}\right)\right) = 1 - \sqrt{3}i

. Its argument is

-\frac{\pi}{3}

.

Why it happens: the expression has all the right ingredients, a modulus and the cosine and sine of one angle, so the sign between them is overlooked.

How to check: convert to algebraic form. Here

b = -2\sin\frac{\pi}{3} = -\sqrt{3}

is negative, so the point lies below the real axis and its argument cannot be

\frac{\pi}{3}

.

Skipping the quadrant check

The angle is taken straight from the arctangent, which only returns values in

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

. For a number in the left half-plane this gives the direction of the opposite point.

Wrong working: for

z = -1 + \sqrt{3}i

,

r = \sqrt{1 + 3} = 2

and

\theta = \arctan\left(\frac{\sqrt{3}}{-1}\right) = -\frac{\pi}{3}

, so

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

. But

2\,\text{cis}\left(-\frac{\pi}{3}\right) = 1 - \sqrt{3}i

.

Right working:

a = -1 < 0

and

b = \sqrt{3} > 0

put

z

in the second quadrant, so add

\pi

:

\theta = -\frac{\pi}{3} + \pi = \frac{2\pi}{3}

, and

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

.

Why it happens:

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

has solutions in two opposite quadrants, and the arctangent always returns the one in the right half-plane.

How to check: convert back.

2\cos\frac{2\pi}{3} = -1

and

2\sin\frac{2\pi}{3} = \sqrt{3}

give the original number; the wrong angle gives

1 - \sqrt{3}i

.

Calculator in the wrong angle mode

An angle given in degrees is entered while the calculator is set to radians, or the other way round. The calculator then evaluates the cosine and sine of a completely different angle.

Wrong working: for

z = 4\,\text{cis}\,60°

in radian mode,

a = 4\cos 60 \approx 4 \cdot (-0.952) \approx -3.81

and

b = 4\sin 60 \approx 4 \cdot (-0.305) \approx -1.22

.

Right working: in degree mode

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

and

b = 4\sin 60° = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3} \approx 3.46

.

Why it happens: the degree sign is easy to overlook, and a calculator keeps whatever mode was last used.

How to check: the signs of

a

and

b

must match the quadrant of the angle. An angle of

60°

lies in the first quadrant, so both parts must be positive; two negative parts point to the wrong mode.

Trigonometric Form FAQ

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 →

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 →

Can $r$ be negative in trigonometric form?

+

No,

r

is a distance, so it is never negative. An expression such as

-2\,\text{cis}\,\frac{\pi}{6}

is a valid number but is not written in trigonometric form: since

-1 = \text{cis}\,\pi

, it equals

2\,\text{cis}\,\frac{7\pi}{6}

, the same angle turned half way round. Reading

-2

as the modulus and

\frac{\pi}{6}

as the argument names the opposite point.Read more →

Is $r(\cos\theta - i\sin\theta)$ in trigonometric form?

+

No, the form needs a plus sign and the same angle in both places. Because cosine is even and sine is odd,

\cos\theta - i\sin\theta = \cos(-\theta) + i\sin(-\theta)

, so the argument is

-\theta

. For example,

2\left(\cos\frac{\pi}{3} - i\sin\frac{\pi}{3}\right) = 1 - \sqrt{3}i

lies below the real axis, with argument

-\frac{\pi}{3}

, not

\frac{\pi}{3}

.Read more →

Why does the principal argument jump on the negative real axis?

+

Because some jump is unavoidable, and the

(-\pi, \pi]

convention puts it there. Follow a point once around the origin and its angle grows by

2\pi

, so any rule that gives one angle per point must drop back by

2\pi

somewhere. With

(-\pi, \pi]

, a point just above the negative real axis has an argument near

\pi

and a point just below it one near

-\pi

; the

[0, 2\pi)

convention moves the break to the positive real axis.Read more →

How does complex multiplication give the angle-addition formulas?

+

Multiply

\text{cis}\,\alpha

by

\text{cis}\,\beta

in two ways. Expanding the brackets with

i^2 = -1

gives the real part

\cos\alpha\cos\beta - \sin\alpha\sin\beta

and the imaginary part

\sin\alpha\cos\beta + \cos\alpha\sin\beta

. The multiplication rule says the product is

\text{cis}(\alpha + \beta)

, with parts

\cos(\alpha + \beta)

and

\sin(\alpha + \beta)

. Matching the parts gives both formulas at once, so one product rebuilds them whenever they are forgotten.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

Converting Trigonometric to Algebraic

Multiplication and Division in Trigonometric Form

Multiplication Rule

Division Rule

Trigonometric Form at a Glance

Common Pitfalls

Trigonometric Form FAQ

Why is the argument not unique?

What is cis notation?

Can rrr be negative in trigonometric form?

Is r(cos⁡θ−isin⁡θ)r(\cos\theta - i\sin\theta)r(cosθ−isinθ) in trigonometric form?

Why does the principal argument jump on the negative real axis?

How does complex multiplication give the angle-addition formulas?

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

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

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

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

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

Can $r$ be negative in trigonometric form?

Is $r(\cos\theta - i\sin\theta)$ in trigonometric form?