Euler’s Formula

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 3e⁵ⁱ in the (a + ib) form using Euler’s formula.

Given: θ = 5

Applying Euler’s formula,

e^iθ = cosθ + isinθ

⟹ e⁵ⁱ = cos5 + i sin5 = 0.284 + i(−0.959) = 0.284 − 0.959i

3e⁵ⁱ = 0.852 – 2.877i

Hence, 3e5i in a + ib is: 

3e⁵ⁱ = 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