Understanding relation of $\sin,\cos,\tan$ with $e$

Question

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?

Answer 1

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 $x\in[0,\pi)$

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

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

Answer 2

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.