## Euler’s Formula

Euler’s variety of complex numbers is crucial enough to earn a separate section. An especially convenient representation that ends up in simplifications in a very lot of calculations. Euler’s complex computation formula is employed to specify the link between trigonometric functions and sophisticated exponential operations.

## About the Topic

Here are two kinds of Euler’s formulas used in additional contexts.

**Euler’s formula for complicated analysis**

e^{iy} = cos y + isin y

**Euler’s formula in the case of polyhedra:**faces + vertices – edges = 2

In this, cos and sin are trigonometric functions, i is the fictitious unit, and e is the base of the log. The performance of this formula will be taken while a complex plane, as a unit complex function eiθ drafting a division circle, where θ could be real and is estimated in radians.

Euler, a Swiss mathematician, gave Euler’s formula. There are two kinds of Euler’s formulas:

**For complicated analysis:**it’s a key direction accustomed to solving complex exponential functions. Euler’s formula is further sometimes called Euler’s equivalence. It does not establish the linkage between trigonometric functions and complex exponential functions.**For polyhedra:**For any polyhedron that doesn’t self-intersect, the quantity of faces, vertices, and edges is expounded in a very particular way, which is given by Euler’s formula or also called Euler’s characteristic.

## Formula

Euler’s formula: e^{iy} = cos y + isin y

Euler’s identity: e^{iπ} +1 = 0

## Solved Examples

**Example 1:** Show e^{i(π/2)} in the (a + ib) form by operating Euler’s method.

As per the question: θ = π/2

Using Euler’s method,

e^{iθ} = cosθ + isinθ

= e^{i(π/2)} = cos(π/2) + isin(π/2)

= 0 + i × 1 = i

Therefore, e^{i(π/2)} in a + ib form is i

**Example 2:** Show 3e^{5i} in the (a + ib) form using Euler’s formula.

Given: θ = 5

Applying Euler’s formula,

e^{iθ} = cosθ + isinθ

⟹ e^{5i} = cos5 + i sin5 = 0.284 + i(−0.959) = 0.284 − 0.959i

3e^{5i} = 0.852 – 2.877i

Hence, 3e5i in a + ib is:

3e^{5i} = 0.852 – 2.877i

**Example 3:** Jeenie learns that a polyhedron has 18 vertices and 28 edges. How can she find the number of faces?

Using Euler’s formula:

F + V − E = 2

F + 18 − 28 = 2

F − 10 = 2

F = 12

Number of faces = 12