B What are cosine and sine functions called in relation to Pi?

Matt Benesi

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1)* What are sine and cosine functions called in relation to Pi?

2) What is the exponential function called in relation to cosine and sine functions?

3) What are the other smooth, continual nested (or iterative) root functions (that are similar to sine and cosine) called in relation to their constants? (see: https://www.physicsforums.com/threads/generating-the-formula-for-the-coefficients-of-an-alternating-series.964892/ )


1* Are they called decomposition functions, since they decompose angles into ratios?
 

mathman

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I have no idea what your questions mean. Could you elaborate?
 

Matt Benesi

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Is there a name for the relationship between Pi and the sine/cosine functions?

Like "Pi is the <xxxxx> of cosine and sine" and "Pi is <xxxx> to sine and cosine". Not just "the period". Is there another name for it?

Extend this to question 2

: "Cosine and sine are the <xxxx> of the exponential function" and "sine and cosine are <xxxx> to the exponential function.
 
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fresh_42

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Is there a name for the relationship between Pi and the sine/cosine functions?

Like "Pi is the <xxxxx> of cosine and sine" and "Pi is <xxxx> to sine and cosine". Not just "the period". Is there another name for it?

Extend this to question 2

: "Cosine and sine are the <xxxx> of the exponential function" and "sine and cosine are <xxxx> to the exponential function.
This doesn't explain anything: a repetition is no explanation.

##\pi## is a zero of the sine function. Otherwise, sine and cosine are just periodic functions. They are related to a circle, which is why ##\pi## plays a role, but it is the circle which is crucial, not ##\pi##.

The real exponential function has nothing to do with the sine and cosine functions. Will say, the complex connection has to be derived, it is not a natural one. Therefore there isn't name for it.

Generally, only things are named, if they are attributed to a person who found them, or if they are frequently used. We use the sine function a lot, but to have ##\pi## as a zero, you already have to make restrictions, e.g. ##\sin(x+1)## hasn't ##\pi## as a zero. Your question doesn't make a lot of sense, which is why you were asked for an explanation, driven by the hope we could answer what really bothers you. If it's only what you wrote, then look for something useful you can name.
 

fresh_42

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It is still pointless. E.g. the periods of sine and cosine can be arbitrary, same as the zeros. Why bound them to ##\pi##? That makes no sense. They are already related via the unit circle and everybody knows this relation. There is no gain in naming the obvious.

The same is true for the formula for ##e^{i \varphi}##, which is called Euler's formula. Why should someone want to rename the exponential function? The name fits far better to its real meaning: exponential growth in nature. Euler's formula is in the end a consequence of integration, namely why the points on the unit circle can be described by it.

To pick a certain formel out of myriads is a deliberate act and does not require names. Names are given if there is a sense behind. I cannot see any sense in the above. It appears to me actually as an advanced version of pointless numerology.
 

Klystron

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Matt Benesi

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@Klystron- there is nothing mystical about numbers, nor the language used to describe them. At least as far as I can tell.

I mentioned the natural log of #2/pi joke, because that's what it is. A joke. Modern English came about after Napier mentioned logarithms, after old Norse "log" became a euphemism for something else. It's not numerology. It's linguistics.

Numerology would be like saying 2^4*3*the 12th prime = the 12th prime *3*4^2, which is why the founding fathers of the USA picked that date. It would be like saying 2*3^2 * the (2*3*2) prime = a number which has digits that in base 10 are the exact numbers of the most common isotope of carbon, which is part of all life. It would be like saying this twelfth prime is the percentage of water's boiling point that human body temperature is. It would be like creating an alphabet, in antiquity, that has an alphanumeric substitution with 2*the 12th prime as Gd, and 2*12 as 24, or the opposite of the answer to life, the universe, and everything, which is 42 (or DB, not bd!).

Or you need Douglas Adams to point out that... well, 4^2*3 * the (3*4*2)th prime =4272. We all know that 24*3=72, which almost was the founding date of the USA (look it up, it was almost 7-2-1776 instead of 7-4). So we better subtract 42*the 12th prime from it to get back to 2718. I am so oblivious sometimes. Sigh.

But that's numerology. And it's sort of interesting, if "pointless" as fresh_42 says. What I was talking about was more... a natural log of number 2 joke. Which.. I think is funny. But I will avoid thinking of math as funny. Because it's not supposed to be funny. It's serious.

Back to a fresh_answer:
It is still pointless. E.g. the periods of sine and cosine can be arbitrary, same as the zeros. Why bound them to ##\pi##? That makes no sense. They are already related via the unit circle and everybody knows this relation. There is no gain in naming the obvious.
Ok, I thought it was obvious that I was talking about evaluating sine and cosine with ℝ values, but I think I need to include more information from question 3 since I'm not interested specifically in the periodic nature of the functions. I was hoping there was some common mathematical name that I wasn't aware of, describing pi and it's relationship to cosine and sine, which would also apply to another thing I am researching.

I am looking for the name of the group of constants and functions that allow one to create continual nested roots, so I can find the generating formula for the terms of the functions (since I already know how to generate the constants for the functions, and extract terms).

So Pi is to [itex]\sqrt{2+sqrt{2+...}}[/itex] like 2.5351045638... is to [itex]\sqrt{.75+\sqrt{.75...}}[/itex]. Pi allows one to smoothly iterate between [itex]\sqrt{2}[/itex] to [itex]\sqrt{2+sqrt{2}}[/itex] to [itex]\sqrt{2+sqrt{2+\sqrt{2}}}[/itex], while ~2.54 allows you to do the same with .75 (given terms of the expansion).

And pi is the only one on the n=2 line that contains every prime, in order, in the functions that extract its #2 roots smoothly. #2. lol.
 

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