### Euler's formula

(formula relating exponentials to trigonometric functions)

Euler's formula is an equation worked out by mathematician Leonhard Euler regarding complex numbers that is heavily depended upon by the applied mathematics of physics, engineering, and related disciplines. It relates exponentials to trigonometric functions, yielding a means to convert an equation of trigonometric functions into one of exponential functions, and vice versa, thus giving the mathematician the choice of the methods of working with a function, such as solving for some variable of interest or calculating with the function. The equation:

```eix = cos x + i sin x
```
• e - Euler's number, a particular irrational number that begins 2.7182...
• i - the square root of minus 1.
• sin, cos - trigonometric functions.
• x - any real number.

The formula was worked out using series expansions, a method of comparing such functions and finding out how they are related, which showed that given the notion of i, this formula produced consistent and useful results. Among the formula's implications is this equation:

```eiπ = -1
or
eiπ + 1 = 0
```

This latter form of this equation is known as Euler's identity.

(mathematics)