Euler's Formula (original) (raw)

Last Updated : 21 Feb, 2026

Euler’s formula establishes a fundamental relationship between exponential functions and trigonometric functions. It shows how complex exponentials can be expressed using sine and cosine.

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

**Where:

The complex exponential e^{i\theta} represents a point on the unit circle in the complex plane, where θ is a real number measured in radians.

**Derivation

Consider the power series expansion of the exponential function:

e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + ...

Substituting x = iθ:

e^{iθ} = 1 + iθ + (iθ)²/2! + (iθ)³/3! + (iθ)⁴/4! + ...

Using i² = −1 and simplifying:

e^{iθ} = 1 + iθ − θ²/2! − iθ³/3! + θ⁴/4! + iθ⁵/5! − ...

Grouping real and imaginary parts:

e^{iθ} = (1 − θ²/2! + θ⁴/4! − ...)

Using the series expansions of cosine and sine:

cosθ = 1 − θ²/2! + θ⁴/4! − ...
sinθ = θ − θ³/3! + θ⁵/5! − ...

Since the real and imaginary parts match the Maclaurin series of cosθ and sinθ respectively, therefore

e^{iθ} = cosθ + i sinθ

**Formula for Polyhedral

Euler’s polyhedral formula states that for any polyhedron that does not self-intersect, the numbers of faces, vertices, and edges are related in a specific way. According to the formula, the sum of the number of faces and vertices is two greater than the number of edges.

F + V - E = 2

where,

**Derivation

Euler's formula can be proven for five platonic solids: cube, tetrahedron, octahedron, dodecahedron and the icosahedron.

**Solids **Number of faces (F) **Number of vertices (V) **Number of edges (E) **F + V - E
**Cube 4 4 6 2
**Tetrahedron 4 6 4 6
**Octahedron 8 6 12 2
**Dodecahedron 12 20 30 2
Icosahedron 20 12 30 2

Applications

Euler’s formula is one of the most important equations in mathematics and has many useful applications. Some of the major applications are:

**Euler’s Identity

Euler’s identity is considered one of the most beautiful equations in mathematics because it connects five fundamental constants in a single simple expression:

e^{i\pi} + 1 = 0

It combines:

Euler’s identity is obtained as a special case of Euler’s formula by substituting θ = π into

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

Since cosπ = −1 and sinπ = 0, we get
e^{i\pi}= -1

which leads to
e^{i\pi} + 1 = 0

**Sample Problems

**Problem 1. Express eiπ/2 in the general form using Euler's formula.

**Solution:

We have,

x = π/2

Using the formula we get,
eix = cos x + i sin x
= cos π/2 + i sin π/2
= 0 + i (1)
= 0 + i

**Problem 2. Express e6i in the general form using Euler's formula.

**Solution:

We have,

x = 6

Using the formula we get,

eix = cos x + i sin x
= cos 6 + i sin 6
= 0.96 + i (-0.279)
= 0.96 - 0.279i

**Problem 3. Express e10i in the general form using Euler's formula.

**Solution:

We have,

x = 10

Using the formula we get,

eix = cos x + i sin x
= cos 10 + i sin 10
= -0.83 + i (-0.544)
= -0.83 - 0.544i

**Problem 4. Express eiπ/3 in the general form using Euler's formula.

**Solution:

We have,

x = π/3

Using the formula we get,

eix = cos x + i sin x
= cos π/3 + i sin π/3
= 0.5 + i (0.86)
= 0.5 + 0.86i

**Problem 5. Verify Euler's formula for a triangular prism.

**Solution:

We have a triangular prism. It is known that,

Number of faces (F) = 5

Number of vertices (V) = 6

Number of edges (E) = 9

Using the formula we have,

F + V − E = 5 + 6 − 9
= 11 - 9
= 2

As the value of F + V − E is 2, the Euler's formula is verified.

**Problem 6. Verify Euler's formula for a square pyramid.

**Solution:

We have a square pyramid. It is known that,

Number of faces (F) = 5

Number of vertices (V) = 5

Number of edges (E) = 8

Using the formula we have,

F + V − E = 5 + 5 − 8
= 10 - 8
= 2

As the value of F + V − E is 2, the Euler's formula is verified.

**Problem 7. Find the number of vertices of a polyhedral if the number of faces is 20 and the number of edges is 30.

**Solution:

We have,

F = 20

E = 30

Using the formula we get,

F + V − E = 2
=> V = E + 2 - F
= 30 + 2 - 20
= 32 - 20
= 12