Some Amazing Applications of Euler’s Formula

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

Keith McNulty


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


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


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

from math import…



Keith McNulty

Pure and Applied Mathematician. LinkedIn Top Voice in Tech. Expert and Author in Data Science and Statistics. Find me on LinkedIn, Twitter or