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How to prove Euler's formula: $\exp(i t)=\cos(t)+i\sin(t)$ ?

I recently heard that $e^{i \pi} = -1$.

WolframAlpha confirmed this for me, however, I don't see how this works.

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marked as duplicate by Qiaochu Yuan May 4 '11 at 22:27

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Do you know the imaginary unit $i$? Do you know how complex exponentiation is defined? –  lentic catachresis May 4 '11 at 22:22
Did you read en.wikipedia.org/wiki/Euler's_formula ? –  lhf May 4 '11 at 22:23
To answer your (now deleted) question on how to put an equation into LaTeX, you can click on edit (below your question) to see what I did. Some links and references on that can be found here –  t.b. May 4 '11 at 22:28
@Theo Buehler, thanks for that. I was trying to use parenthesis instead of braces. I'll remember that next time. –  mbreedlove May 5 '11 at 0:24

2 Answers 2

up vote 12 down vote accepted

This identity follows from Euler's Theorem, \begin{align} e^{i \theta} = \cos \theta + i \sin \theta, \end{align} which has many proofs. The one that I like the most is the following (sketched). Define $f(\theta) = e^{-i \theta}(\cos \theta + i \sin \theta)$. Use the quotient rule to show that $f^{\prime}(\theta)= 0$, so $f(\theta)$ is constant in $\theta$. Evaluate $f(0)$ to prove that $f(\theta) = f(0)$ everywhere.

Take $\theta = \pi$ for your claim.

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