# Euler’s Formula

Euler’s formula holds a prominent place in the field of mathematics. It aids in establishing the essential link between trigonometric functions and complex exponential functions. It is a crucial formula used for solving complicated exponential functions. It is also known as Euler’s identity. It has a lot of applications in complex analysis and is used to find the number of vertices and faces of a polyhedral.

**Formula for Complex Analysis**

The complex analysis formula provides us with a way to express the imaginary power of the exponential functions in form of trigonometric ratios. Its formula can be related to a complex plane, where a unit complex function e^{ix} traces a unit circle, such that x is a real number measured in radians.

e^{ix}= cos x + i sin xwhere,

x is a real number,

e is the logarithm’s base,

sin x and cos x are the trigonometric functions,

i is the imaginary part.

The expression cos x + i sin x is also known as polar form of a complex number.

**Derivation**

Consider the expansion series of the exponential function e

^{x}.e

^{x}= 1 + x + x^{2}/2! + x^{3}/3! + x^{4}/4! + …….. + ∞Substituting x as ix we get,

e

^{ix}= 1 + ix + (ix)^{2}/2! + (ix)^{3}/3! + (ix)^{4}/4! + …….. + ∞Using the property i

^{2}= -1 we get,= 1 + ix – x

^{2}/2! – ix^{3}/3! + x^{4}/4! + …….. + ∞= (1 – x

^{2}/2! + x^{4}/4! – …….. + ∞) + i (x – x^{3}/3! + x^{5}/5! – …….. + ∞)Using series expansion property of sine and cosine functions we get,

= cos x + i sin x

This derives the Euler’s formula for complex analysis.

**Formula for Polyhedral**

The polyhedral formula of Euler states that the number of faces, vertices, and edges of every polyhedron that does not self-intersect are connected in a certain way. Its formula says that the number of vertices and faces of a polyhedral combined is two greater than its number of edges.

F + V – E = 2where,

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

Cube4

4

6

2

Tetrahedron6

8

12

2

Octahedron8

6

12

2

Dodecahedron12

20

30

2

Icosahedron20

12

30

2

**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 e ^{6i} 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 e ^{10i} 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 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

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