Linked Questions

67
votes
16answers
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 ...
69
votes
9answers
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 ...
34
votes
11answers
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 ...
46
votes
10answers
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 ...
23
votes
2answers
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.
1
vote
4answers
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 ...
3
votes
6answers
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 ...
4
votes
3answers
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 ...
2
votes
2answers
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:
1
vote
2answers
152 views

How does Taylor series work for sine and cosine?

I was just wondering if anyone can help me understand taylor series for sine and cosine. I have no background in calculus but I always found it interesting how the ratios of the arc to the radius was ...
0
votes
1answer
129 views

Complex integration and differentiation for $\gamma:[a,b]\rightarrow \mathbb{C}$

Let $\gamma(t):[a,b]\rightarrow \mathbb{C}$ , $[a,b]\subset \mathbb{R}$, $\gamma(t)= \gamma_1(t) + i\gamma_2(t)$ $\int_{a}^{b} \gamma(t) dt = \int_{a}^{b} \gamma_1(t) dt + i\int_{a}^{b} \gamma_2(t) ...
0
votes
1answer
53 views

Can $\lim_{h\to 0} (1+h)^{[\frac{x}{h}]}$ be an equivalent definition of the exponential function?

Here, take $[x]$ to be the smallest integer function. I like this more than $\lim_{n\to \infty} (1+\frac{x}{n})^n$ because it seems to make the property $\exp(x+y)=\exp(x)\exp(y)$ obvious. This is ...