Euler’s formula establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler’s formula or Euler’s equation is one of the most fundamental equations in maths and engineering and has many applications. However, there exist two Euler’s formulas in which one is for complex analysis and the other for polyhedrons.
Let’s have a look at the formula in each case and the solved example.
Euler’s Formula Equation
Euler’s formula or Euler’s identity states that for any real number x, in complex analysis is given by:
| eix = cos x + i sin x |
Where,
- x = real number
- e = base of natural logarithm
- sin x & cos x = trigonometric functions
- i = imaginary unit
Note: The expression cos x + i sin x is often referred to as cis x.
Example Question Using Euler’s Equation Formula
Question: Find the value of e i π/2.
Solution:
Given ei π/2
Using Euler’s formula,
eix = cos x + i sin x
e i π/2 = cos π/2 + i sin π/2
e i π/2 = 0 + i × 1
e i π/2 = i
Euler’s Formula for Polyhedrons
Euler’s polyhedra formula shows that the number of vertices and faces together is exactly two more than the number of edges. We can write Euler’s formula for a polyhedron as:
Faces + Vertices = Edges + 2
F + V = E + 2
Or
F + V – E = 2
Here,
F = number of faces
V = number of vertices
E = number of edges
Let us verify this formula for some solids.
| Solid name | Faces (F) | Vertices (V) | Edges (E) | Result (F + V – E) |
| Cube | 6 | 8 | 12 | 6 + 8 – 12 = 2 |
| Square pyramid | 5 | 5 | 8 | 5 + 5 – 8 = 2 |
| Triangular prism | 5 | 6 | 9 | 5 + 6 – 9 = 2 |
Example on Euler’s Formula for Solids
Example: If a polyhedron contains 12 faces and 30 edges, then identify the name of the polyhedron.
Solution:
Given,
Number of faces = F = 12
Number of edges = 30
Using Euler’s formula of solids,
F + V = E + 2
12 + V = 30 + 2
V = 32 – 12
V = 20
From the above values of F, V and E, we can say that the polyhedron could be Dodecahedron.
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