What is Euler's identity?

Euler's identity is the special case e^(iπ) + 1 = 0, obtained by setting θ = π in Euler's formula. It links five fundamental constants — e, i, π, 1, and 0 — in a single equation and is often called the most beautiful formula in mathematics.

How do you use this Euler's formula explorer?

Drag the blue point on the complex plane or use the θ slider to change the angle and the r slider to change the radius. The tool displays a right triangle, live cosine and sine values, and a step-by-step formula breakdown that updates in real time.

Why does the triangle disappear at certain angles?

At θ = 0, π, π/2, and 3π/2 the triangle collapses because one trigonometric component equals zero. At 0 and π, sin θ = 0 so there is no vertical leg. At π/2 and 3π/2, cos θ = 0 so there is no horizontal leg. These are the axis-crossing degenerate cases.

How does Euler's formula relate to polar form?

Polar form writes any complex number as z = r·e^(iθ), where r is the modulus (distance from origin) and θ is the argument (angle). Euler's formula provides the bridge: r·e^(iθ) = r cos θ + i·r sin θ, converting between polar and rectangular representations.

Euler's Formula Explorer

e^iθ = cos θ + i sin θ

θ45.0° = π/4

Live Values

e^iθ=0.707 + 0.707i

θπ/4 = 45.0°cos θ0.707sin θ0.707Re(z)0.707Im(z)0.707|z|1

e^(iπ/4)

45° — cos = sin = √2/2 ≈ 0.707

Formula Breakdown

1Start with Euler's formula: e^iθ = cos θ + i sin θ

2Substitute θ = π/4:

ei·π/4 = cos(π/4) + i sin(π/4)

3Evaluate:

= 0.707 + i · 0.707 = 0.707 + 0.707i

Key Ideas

e^iθ traces the unit circle. As θ goes from 0 to 2π, the point completes one full revolution. The real part is cos θ, the imaginary part is sin θ.

θ is the angle from the positive real axis, measured counterclockwise in radians. One full turn = 2π radians = 360°.

The right triangle connects trig to complex numbers. The hypotenuse is r (the modulus), the horizontal leg is r cos θ (real part), and the vertical leg is r sin θ (imaginary part).

Why use e^iθ instead of cos θ + i sin θ? Because multiplication becomes simple: e^iα · e^iβ = e^i(α+β). Multiplying complex numbers = adding angles. The exponential form makes rotation algebra trivial.

r · e^iθ is the polar form of any complex number. r is the distance from the origin (modulus), θ is the angle (argument). Every complex number can be written this way.

Getting Started — Drag the Point

Navigating All Four Quadrants

Landmark Angle Presets

Degenerate States — When the Triangle Collapses

Adjusting the Radius

Reading the Live Values and Formula Breakdown

What is Euler's Formula?

Euler's Identity — The Special Case at θ = π

The Right Triangle, Trigonometry, and Polar Form

Related Concepts and Tools

Getting Started — Drag the Point

The blue draggable point on the complex plane represents the value of

re^{i\theta}

. Grab it and move it anywhere within the plane to explore how Euler's formula connects angles, trigonometry, and complex numbers in real time.

As you drag, the right panel updates instantly. You will see the current angle

\theta

in both degrees and radians, the cosine and sine values, and the resulting complex number in rectangular form. A colored right triangle appears connecting the origin to your point, with the horizontal leg showing the real part and the vertical leg showing the imaginary part.

Start by dragging the point slowly around the unit circle. Watch how the triangle changes shape, how the projections on both axes shift, and how the formula breakdown at the right walks through each substitution step. Every position you place the point produces a unique geometric snapshot of Euler's formula in action.

Navigating All Four Quadrants

Each quadrant of the complex plane produces a visually distinct triangle with different sign combinations for cosine and sine. Drag the point into each quadrant to see how the triangle flips and the values change sign.

Quadrant I (upper right): both

\cos\theta > 0

and

\sin\theta > 0

. The triangle sits in the standard position with the green horizontal leg pointing right and the red vertical leg pointing up.

Quadrant II (upper left):

\cos\theta < 0

while

\sin\theta > 0

. The horizontal leg now extends to the left of the imaginary axis. The real part of

e^{i\theta}

becomes negative.

Quadrant III (lower left): both components are negative. The triangle appears below and to the left of the origin, and the complex number lies in the third quadrant.

Quadrant IV (lower right):

\cos\theta > 0

while

\sin\theta < 0

. The vertical leg drops below the real axis. This corresponds to angles between

\frac{3\pi}{2}

and

2\pi

.

Each quadrant configuration makes a distinct illustration showing how the signs of the real and imaginary parts depend on where the angle places the point.

Landmark Angle Presets

Seven preset buttons below the sliders snap the explorer to important angles on the unit circle:

0

\frac{\pi}{6}

\frac{\pi}{4}

\frac{\pi}{3}

\frac{\pi}{2}

\pi

, and

\frac{3\pi}{2}

. Each button also resets the radius to

r = 1

, placing the point exactly on the unit circle.

Click

\frac{\pi}{6}

(30°) to see the classic 30-60-90 triangle with

\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2} \approx 0.866

and

\sin\frac{\pi}{6} = \frac{1}{2} = 0.5

. The horizontal leg is noticeably longer than the vertical one.

Click

\frac{\pi}{4}

(45°) and the triangle becomes isosceles — both legs have equal length since

\cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2} \approx 0.707

. The point sits exactly on the diagonal.

Click

\frac{\pi}{3}

(60°) and the triangle mirrors the 30° case: now the vertical leg is longer. Together, these three angles illustrate how the balance between real and imaginary parts shifts as

\theta

increases through the first quadrant.

Click

\pi

to see Euler's identity in action — the point lands at

-1

on the real axis, confirming that

e^{i\pi} = -1

Degenerate States — When the Triangle Collapses

At certain angles the right triangle collapses into a line segment because one of the two trigonometric components equals zero. These degenerate configurations are important special cases of Euler's formula.

At

\theta = 0

: the point sits at

(r, 0)

on the positive real axis. Since

\sin 0 = 0

, the vertical leg vanishes entirely and the triangle reduces to a horizontal line. The formula reads

e^{i \cdot 0} = 1

.

At

\theta = \frac{\pi}{2}

: the point lands at

(0, r)

on the positive imaginary axis. Now

\cos\frac{\pi}{2} = 0

, so the horizontal leg disappears. Only the vertical red segment remains. This gives

e^{i\pi/2} = i

, a purely imaginary result.

At

\theta = \pi

: the point reaches

(-r, 0)

on the negative real axis — another horizontal-only state. The formula yields the famous

e^{i\pi} = -1

.

At

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

: the point drops to

(0, -r)

on the negative imaginary axis, producing a downward vertical segment. Here

e^{i3\pi/2} = -i

.

These four states correspond to the axis crossings of the unit circle. Each one produces a clean, degenerate illustration with no triangle — just a single colored line along one axis.

Adjusting the Radius

The

r

slider controls the modulus (distance from the origin) and ranges from

0.1

2.4

. When

r = 1

, the point lies on the solid unit circle. When

r \neq 1

, a dashed circle appears at radius

r

, and the triangle labels switch from "

\cos\theta

" / "

\sin\theta

" to "

r\cos\theta

" / "

r\sin\theta

".

Try setting

\theta = \frac{\pi}{4}

and then slowly increasing

r

from

1

2

. The triangle grows proportionally — its shape stays the same because the angle has not changed, but every side length doubles. The live values panel reflects the scaled components: at

r = 2

, the real and imaginary parts are both

2 \times 0.707 \approx 1.414

.

Setting

r

below

1

shrinks the triangle inside the unit circle. At

r = 0.5

, the point sits halfway to the unit circle and all component values are halved.

This demonstrates the general polar form

z = re^{i\theta}

, where

r

scales the unit-circle point outward or inward. The angle determines direction; the radius determines magnitude.

Reading the Live Values and Formula Breakdown

The right panel provides two complementary readouts that update with every change to

\theta

r

.

The Live Values section displays six quantities: the current angle

\theta

in radians and degrees,

\cos\theta

(green, matching the horizontal leg),

\sin\theta

(red, matching the vertical leg),

\text{Re}(z)

and

\text{Im}(z)

as the rectangular coordinates, and

|z|

as the modulus. When

r = 1

, the real and imaginary parts equal

\cos\theta

and

\sin\theta

directly.

The Formula Breakdown walks through the substitution step by step. Step 1 states Euler's formula. Step 2 plugs in the current

\theta

value. Step 3 evaluates cosine and sine numerically and displays the final complex number. When

r \neq 1

, an additional multiplication step appears showing

r \cdot e^{i\theta} = r\cos\theta + ir\sin\theta

.

At landmark angles, an orange callout box appears with the symbolic result — for example, "

e^{i\pi} = -1

" and a note explaining its significance. This callout only activates when

r = 1

and the angle is within a small tolerance of a preset value.

What is Euler's Formula?

Euler's formula states that for any real number

\theta

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

This equation bridges three seemingly unrelated mathematical objects: the exponential function, trigonometric functions, and the imaginary unit

i

. It reveals that raising

e

to an imaginary power produces a point on the unit circle in the complex plane, with the angle

\theta

measured in radians from the positive real axis.

The formula can be derived from the Taylor series expansions of

e^x

\cos x

, and

\sin x

. When

x = i\theta

is substituted into the exponential series, the real terms collect into the cosine series and the imaginary terms collect into the sine series.

This is one of the most important results in mathematics because it unifies algebra, geometry, and analysis. It converts between rectangular form

a + bi

and polar form

re^{i\theta}

, making operations like multiplication, division, and exponentiation of complex numbers far simpler.

Euler's Identity — The Special Case at θ = π

Setting

\theta = \pi

in Euler's formula gives:

e^{i\pi} = \cos\pi + i\sin\pi = -1 + 0i = -1

Rearranging:

e^{i\pi} + 1 = 0

. This is Euler's identity, often called the most beautiful equation in mathematics because it links five fundamental constants —

e

i

\pi

1

, and

0

— in a single compact relation.

In the explorer, click the

\pi

button to see this visually. The point lands at

(-1, 0)

on the negative real axis. The triangle collapses to a horizontal line pointing left, and the orange landmark callout confirms the identity.

Similarly, setting

\theta = \frac{\pi}{2}

gives

e^{i\pi/2} = i

, meaning that multiplying by

e^{i\pi/2}

rotates any complex number by 90° counterclockwise. And

\theta = 2\pi

returns to

e^{i \cdot 2\pi} = 1

, completing a full revolution. These special cases demonstrate that the exponential function naturally encodes rotation in the complex plane.

The Right Triangle, Trigonometry, and Polar Form

The colored right triangle displayed in the explorer is the geometric heart of Euler's formula. Its three sides directly represent the three parts of the equation

re^{i\theta} = r\cos\theta + ir\sin\theta

.

The navy hypotenuse from the origin to the point

z

has length

r = |z|

, the modulus. The green horizontal leg from the origin to the projection on the real axis has length

|r\cos\theta|

, the real part. The red vertical leg from the real-axis projection up to

z

has length

|r\sin\theta|

, the imaginary part.

This is identical to the standard trigonometric relationship in a right triangle where the adjacent side is

r\cos\theta

and the opposite side is

r\sin\theta

. The formula

e^{i\theta}

simply packages this triangle into exponential notation.

The polar form

z = re^{i\theta}

is useful because multiplication of complex numbers becomes:

z_1 \cdot z_2 = r_1 r_2 \cdot e^{i(\theta_1 + \theta_2)}

Moduli multiply, angles add. This is far simpler than expanding

(a + bi)(c + di)

in rectangular form. The explorer makes this visible: the angle

\theta

controls rotation while

r

controls scaling.

Related Concepts and Tools

Euler's formula connects to many areas of complex number theory and applied mathematics. Explore these related topics to deepen your understanding.

Complex Number Explorer — an interactive tool for visualizing complex arithmetic, plotting numbers in rectangular and polar form, and performing operations on the complex plane.

Complex Numbers — foundational theory covering the imaginary unit

i

, rectangular form

a + bi

, and algebraic operations.

Polar Form and Modulus — detailed coverage of writing complex numbers as

re^{i\theta}

, computing the modulus

|z|

and argument

\arg(z)

.

De Moivre's Theorem — extends Euler's formula to integer powers:

(e^{i\theta})^n = e^{in\theta}

, which gives

( \cos\theta + i\sin\theta )^n = \cos(n\theta) + i\sin(n\theta)

.

Roots of Unity — the

n

-th roots of

1

are

e^{i \cdot 2\pi k/n}

for

k = 0, 1, \dots, n-1

, equally spaced around the unit circle.

Trigonometric Identities — Euler's formula provides elegant proofs of angle-sum, double-angle, and product-to-sum identities by manipulating exponentials.