\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Precalculus - Euler's Formula

Single Variable Calculus
Multi-Variable Calculus

Euler's Formula

\(e^{i\theta} = \cos(\theta)+i\sin(\theta)\)

Euler's Formula is where exponentials, complex numbers and trig all combine into one elegant formula, \(e^{i\theta} = \cos(\theta)+i\sin(\theta)\). This may not seem very useful but in electronics, for example, we convert trig functions to complex exponental form, using this equation, to make calculations easier. But how do we do that?

The idea is replace \(\theta\) with \(-\theta\) to get \(e^{-i\theta} = \cos(-\theta)+i\sin(-\theta) = \cos(\theta)-i\sin(\theta)\). Then we add this to \(e^{i\theta}\) and solve for cosine. Similarly, we subtract them and solve for sine. This process yields these equations.

\(\displaystyle{ \cos(\theta) = \frac{e^{i\theta}+e^{-i\theta}}{2} }\)

\(\displaystyle{ \sin(\theta) = \frac{e^{i\theta}-e^{-i\theta}}{2i} }\)

Rather than just taking Euler's Formula for granted, let's look closely at what it is telling us. Here is a great video giving an intuitive explanation of the equation and what it means. This guy is great. He takes complicated concepts and makes them easier to see and understand.

BetterExplained - Understanding Euler's Formula [9min-53secs]

video by BetterExplained

Now all this may seem a bit overwhelming but let's watch a great video explaining more intuitively why \(e^{i\pi}=-1\) holds.

3Blue1Brown - Understanding e to the pi i [6min-13secs]

video by 3Blue1Brown

Proofs of Euler's Formula

These proofs require calculus.

Derek Owens - Calculus 6.11 - Euler's Identity (using calculus) [12min-25secs]

video by Derek Owens

Dave Academy - Euler's Formula Proof (Taylor Series) [10min-26secs]

video by Dave Academy

Dr Chris Tisdell - Euler's formula: A cool proof (using differential equations) [11min-55secs]

video by Dr Chris Tisdell

Really UNDERSTAND Precalculus

Log in to rate this page and to see it's current rating.

Topics You Need To Understand For This Page

complex numbers


irrational number \(e\)

To bookmark this page, log in to your account or set up a free account.

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

calculus motivation - music and learning

Shop Amazon - Used Textbooks - Save up to 90%

As an Amazon Associate I earn from qualifying purchases.

I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me.

Support 17Calculus on Patreon


Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.


Copyright © 2010-2022 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

Real Time Web Analytics