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Let us start with the obvious.

I know the formulae for angles. I know how to apply them. I also know the formulae involving $e$.

But I don't understand what sine has to do with Euler's $e$. (Neither do I for cosine or tangent)

If you were to build a course that relies on truly understanding those three functions and to a certain degree their implications, where would you start?

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    Are you familiar with Taylor series? – 2017-01-22
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    @MichaelMcGovern I heard (i.e. can google it) of it, but that is out of my reach at the moment – 2017-01-22
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    It is a bit hard to understand exactly what you're asking. However, if you want proof of Euler's formula Taylor series are one of the classic ways of demonstrating that – 2017-01-22
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    Lol sinus does not mean what I think you think it means – 2017-01-22
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    Lol. Hello and welcome to the link between geometry and algebra. – 2017-01-22
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    I invite you to illuminate me(and maybe others at the chat): http://chat.stackexchange.com/rooms/36/mathematics . However, if you feel this question is too broad, feel free to say so. – 2017-01-22
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    http://math.stackexchange.com/questions/3510/how-to-prove-eulers-formula-eit-cos-t-i-sin-t – 2017-01-22
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    @TehRod: I bet the OP's native language isn't English. Nor is it my native language. That's how I know that in some languages "sinus" means "sine" (and "sinusit" or something like that means "sinus"). They do come from the same Latin root, so it is pretty much the same word! :-) – 2017-01-22
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    @zipirovich hmm I did not know that, that is actually quite interesting thank you – 2017-01-22
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    @TehRod: You're welcome! Needless to say, the names "cosinus" and "tangens" also must be the names of these functions in the OP's native language, as they are in mine. – 2017-01-22
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    [Here](http://math.stackexchange.com/revisions/670237/1) are [some](http://math.stackexchange.com/revisions/727300/1) intuitive [insights](http://math.stackexchange.com/revisions/1222834/1). – 2017-01-25

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Well, we need a definition for $e^x$, which is the only part your missing. And...I choose this one!

$$e^x=\lim_{n\to\infty}\left(1+\frac xn\right)^n$$

And it just so happens that if I let $x\to ix$, I get the following:

$$e^{ix}=\lim_{n\to\infty}\left(1+\frac{ix}n\right)^n$$

An animation of this for $x\in[0,\pi)$

enter image description here

Interestingly, it approaches a circle, which, if we remember the Cartesian coordinates on the unit circle:

$$e^{ix}=\cos(x)+i\sin(x)$$

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    Note that the animation, specifically, is for $x=\pi$. – 2017-01-23
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    @AkivaWeinberger well, it was borrowed from a Wikipedia purely about $e^{i\pi}+1=0$. – 2017-01-23
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    In order to illustrate a trajectory as a function of $t,$ you have to start plotting the trajectory somewhere and stop somewhere. Starting at $t=0$ and stopping when you reach the estimated position at $t=\pi$ is a happy choice even if your main purpose is not to prove Euler's identity. The animation illustrates how the change in radius along the trajectory goes to zero as $n\to\infty.$ – 2017-02-25
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    You are implicitly using the fact that the limit definition of $e^{ix}$ leads to your last equation. The proof although not difficult is non-trivial. You can have a look at http://math.stackexchange.com/a/2161021/72031 or http://math.stackexchange.com/a/1668179/72031 – 2017-02-25
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    @ParamanandSingh Thanks. When looking at this problem, I suppose I trade triviality for intuition. – 2017-02-25
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It's a bit of a vague question. Euler's is usually the relation we think of when relating trig functions to $e$. It states:

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

The typical proof involves Taylor Series. If you don't know about Taylor Series an "easier" way is to prove it:

  1. Verify the initial conditions are the same.
  2. Show that both $e^{i\theta}$ and $\cos(\theta) + i\sin(\theta)$ satisfy:

$$D_zf(z) = if(z)$$

The full proof can be found here. But you should try working it out yourself.

A more geometric interpretation (however, not really a proof) is that both $e^{i\theta}$ and $\cos(\theta) + i\sin(\theta)$ both represent the circle on the complex plane.

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    Not sure how they'd both represent a circle on the complex plane if you haven't defined $e^{i\theta}$ yet. – 2017-01-22
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    @SimplyBeautifulArt It's not part of the proof. Just intuition. The formal proof is in the first part. – 2017-01-22
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    @SimplyBeautifulArt To put another way, you prove Euler's formula, and then you notice that $\cos(\theta) + i\sin(\theta)$ is a circle, and thus $e^{i\theta}$ describes a circle in the complex plane. – 2017-01-22
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    I don't want to speak down to this post. it was probably in the best intentions. But for me truly understanding something is being able to imagine it. But maybe i make up my mind in the future and see the value in your answer. Don't be deterred by my ignorance ;). I want to add that I upvoted, just not accepted this as an answer. – 2017-01-22