Euler's Formula

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:

• θ is a real number (in radians)
• e is the base of the natural logarithm
• sinθ and cosθ are trigonometric functions
• i is the imaginary unit (i² = −1)

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! − ...)

• i(θ − θ³/3! + θ⁵/5! − ...)

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,

• F is the number of faces,
• V the number of vertices,
• E the number of edges.

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:

• Deriving the famous Euler’s identity
• Writing complex numbers in exponential form
• Providing alternative definitions of trigonometric and hyperbolic functions
• Extending exponential and logarithmic functions to complex numbers
• Giving alternative proofs of De Moivre’s theorem and trigonometric addition formulas.

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:

• 0 — additive identity
• 1 — unity
• π — circle constant
• e — base of the natural logarithm
• i — imaginary unit (i² = −1)

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 e^iπ/2 in the general form using Euler's formula.

Solution:

We have,

x = π/2

Using the formula we get,
e^ix = 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,

e^ix = 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,

e^ix = cos x + i sin x
= cos 10 + i sin 10
= -0.83 + i (-0.544)
= -0.83 - 0.544i

Problem 4. Express e^iπ/3 in the general form using Euler's formula.

Solution:

We have,

x = π/3

Using the formula we get,

e^ix = 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