13k views

### Why is radian so common in maths?

I have learned about the correspondence of radians and degrees so 360° degrees equals $2\pi$ radians. Now we mostly use radians (integrals and so on) My question: Is it just mathematical convention ...
38k views

### Why is $\pi$ equal to $3.14159…$?

Wait before you dismiss this as a crank question :) A friend of mine teaches school kids, and the book she uses states something to the following effect: If you divide the circumference of any ...
5k views

### How do I interpret Euler's formula? [duplicate]

I don't understand the formula at all: $$e^{ix} = \cos(x) + i \sin(x)$$ I've tried reading all sorts of webpages and answers on the subject but it's just not clicking with me. I don't understand how ...
33k views

### Why do we require radians in calculus?

I think this is just something I've grown used to but can't remember any proof. When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does ...
1k views

### How to prove this $\pi$ formula? [duplicate]

I am hoping to find out where the formula $$\frac{\pi}{2}=\sum_{k=0}^{\infty}\frac{k!}{\left(2k+1\right)!!}$$ comes from. I can't see how one could begin to prove it.
3k views

### Proof for hyperbolic trigonometric identities [closed]

I've been studying hyperbolic functions and was wondering where the following two identities were derived from: $$\sinh(x) = \frac{e^{x}-e^{-x}}{2}$$ $$\cosh(x) = \frac{e^{x}+e^{-x}}{2}$$ I ...
285 views

### Intuitive explanation of $y' = y \implies y = Ce^x$

I understand why $f : \mathbb{R} \to \mathbb{R}$ with $f'(x) = f(x)$ and $f(0) = 1$ must be $f (x) = e^x$, but I don't really feel it is super intuitive. Intuitively, why would you expect such a ...
212 views

### Why should one be fascinated with $e^{i \pi} +1 = 0$? [closed]

I know some people say that $e^{i \pi} +1= 0$ is cool because it links fundamental yet disparate constants. But to me, the existence of a "nice" equation like this is not surprising. For hundreds of ...
144 views

### Finding $\frac {d(\tan \theta)}{d\theta}$

I was reading Needham's Visual Complex Analysis, and could not figure out how we get length $Ld\theta$ here: