# Some Amazing Applications of Euler’s Formula

## Euler’s simple formula has so many cool applications in trigonometry

Euler’s formula is a simple expression that links complex numbers with trigonometry. It goes like this:

cosθ is thus the *real part* of the identity and sinθ is the *imaginary part.*

Using Euler’s formula, it’s pretty straightforward to express standard trig ratios in terms of complex exponents, for example, noting that cos(-θ) = cosθ and sin(-θ) = -sinθ, we can say that

and

## Deriving well-known trigonometric identities

This way of expressing standard trig ratios offers us the possibility to use Euler’s formula to prove certain standard trig identities, for example:

Rearranging this we have:

Using an alternative manipulation we can determine:

So that:

Subtracting or adding these two identities tells us the well-known high school trig identities:

And similar manipulation can be done for sin(⍺+β) and sin(⍺-β) to find their associated identities.

## Simplifying trigonometric series

Euler’s formula can also be used to efficiently and effectively sum series involving trigonometric expressions. Consider the following expression:

Using Euler’s formula, we can say that this is the real part of an exponent sum as follows (using Re() to denote ‘the real part’):

Some further manipulation gives us:

Now notice the standard geometric series with first term 1, *n+1 *terms and common ratio e^{i2β}. Therefore, we can now say that our sum is

Therefore:

Let’s verify this using a couple of examples in Python:

`from math import…`