This short note is intended to draw attention to one rather unobvious property of trigonometric functions, namely, the dependence on the metric in which we work. Under the cut, I cannot promise the strictest mathematical calculations and generally accepted terminology, but what I promise is that there are many pictures that will saturate your inquisitive mind with an understanding of alternative trigonometry.

Small but damn interesting preamble

The idea for this article came to me by chance, when I once again pounced on the notorious equation

$$

$$cos(x)=x$$

In an attempt to solve it ANALYTICALLY, that is, to get a so-called closed-form solution or an answer in the form of a finite composition of numbers (something like $$$\sqrt 5 /2 + 17\pi$). By the way, try it yourself! After dropping 15 minutes over this equation, you will quickly understand that the school knowledge provided to you is clearly not enough (probably) to solve this equation, and complex Vyshmatov's tricks make the solution completely unattainable. After your first surrender to this equation, you will want to go to Google, and he will give you a Dottie number with appropriate impartiality , offering a compromise solution in the form of an infinite converging numerical sequence.

By the way, the numerical solution: $$$x=0.739085...$...

, , , , talis qualis. , , , - ( ).

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— , , . , :

$$

$$c=\sqrt{a^2+b^2}$$

. . , . , .

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$$

$$c=|a+b|$$

:

$$

$$\sin(x)=\frac{a}{|a+b|},\ \cos(x)=\frac{b}{|a+b|}$$

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, ( : ). ( ). : 1, $$$L_p$ , , ( ):

$$

$$(a^p+b^p)^\frac{1}{p}=1$$

p , :

, $$$p=2$ , $$$p=1$ — . , p ( ), … .

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$$

$$\sin_p(x),\ \cos_p(x),\ \tan_p(x),$$

$$$p$ . , : $$$\sin_2(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$ $$$\tan_2(\frac{\pi}{4})=1$. , , $$$y=kb$, $$$k$ — , $$$b$ — (. ).

, , , :

1. $$$x$ $$$\tan_2(x)$;
2. : $$$k=\tan_2(x)$, .
3. $$$kb=|1-b|$ $$$b\in[0,1]$.
4. $$$b$ $$$a=|1-b|$
5. $$$\sin_1(x),\ \cos_1(x)$ ( ) .

, $$$L_p$ .

$$

$$\sin_p(x)=\frac{a}{(a^p+b^p)^{1/p}},\ \cos_p(x)=\frac{b}{(a^p+b^p)^{1/p}}$$

, ? ! .

— , : $$$x\in[0,\pi/2]$. "" .

, ( "not concaveness") . : " ?".

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. programming language built-in function Cos (), remember that you are doomed to receive in response only legal, time-tested, loved by everyone $$$\cos_2()$... And only those who decided to take a desperate step to entercustom underground, they will find something new and surprising for themselves).

Sorry to leave the footer without links to any sources on this topic. 10 minutes of honest googling in English did not lead me to anything similar to habr or anywhere else. I would be happy to add this to the update from the comments.

For those who like to look into the code