Defined by and named after **Leonhard Euler**, Euler’s Formula not only changed the way mathematicians analyze functions, polar coordinates, and cartesian coordinates, but it also paved the way for arguably, mathematic’s most beautiful equation: Euler’s Identity.

**Euler’s Identity**

Termed by Mathematicians as the **most beautiful formula in Mathematics**, Euler’s Identity takes in irrational values like **π** and **e** alongside non-real values like **i** (Imaginary Number) and outputs -1. When 1 is added to the value, then it returns 0, thus forming a very visually and mathematically appealing function.

This aesthetic equation is an instance of **Euler’s Formula**, where x (Input Variable) = 1

**Euler’s Formula**: e^(ix) = cos(x) + isin(x)

Euler’s Formula is a very important function when converting from Cartesian to Polar Coordinates and vice-versa. (See the Parametric Equations and Polar Curves Sections for more information) It notably expresses a mathematical relationship between the trigonometric functions cosx and sinx with exponential functions.

## But how does this work? -> Derivation of Euler’s Formula

Great question! As noted earlier, Euler’s Identity is just an instance of Euler’s Formula. Euler’s Formula, moreover, can be derived using Maclaurin/Taylor Series expansions! Here is how we can derive Euler’s Formula…