Can 2(sinx) be called a 'trigonometric function'?

[Psi137]

TL;DR Summary

trigonometric function, composite function

Recently I debated with a friend of mine, and the topic was 'Is it reasonable that we call 2(sinx) as trigonometric function?'.
My friend said that if we can call y=2(sinx) as trigonometric function, we can call y=x as trigonometric too, because if we call the former 'correct', there is no reason that we shouldn't call y=arcsin(sinx) as trigonometric function.
Generally, in encyclopedia, it defines trigonometric function for only 6 types, sin, cos, tan, csc, sec, cot.
Then, here is total conclusion :
If y=f(x) is trigonometric function, can y=g(f(x)) be called trigonometric function?

 

[BvU]

Hi $\psi , \quad$ !

My friend said that if we can call y=2(sinx) as trigonometric function, we can call y=x as trigonometric too, because if we call the former 'correct', there is no reason that we shouldn't call y=arcsin(sinx) as trigonometric function.

The mistake here is the assumption that the two functions are equivalent. They are not.

I support the 'total conclusion'.

 

[PeroK]

Summary:: trigonometric function, composite function

Recently I debated with a friend of mine, and the topic was 'Is it reasonable that we call 2(sinx) as trigonometric function?'.

If you are trying to learn mathematics a debate like that is a waste of time.

 

[Psi137]

If you are trying to learn mathematics a debate like that is a waste of time.

Yeah, of course I know this is of no use, but I need the answer of this junk so that I can withdraw or revise my thinking

 

[BvU]

I need the answer of this junk

They are not.

Confirm this and you are in business

 

[Psi137]

The mistake here is the assumption that the two functions are equivalent. They are not.

I support the 'total conclusion'.

Can you tell me why that mistake happens and why you supported it?

Thank you.

 

[jedishrfu]

Here’s a dictionary reference

https://www.thefreedictionary.com/trigonometric+function

where you can see some variation in the definition of these functions.

Sometimes,, in life the only way to win the game is not to play at all. (paraphrased from Wargames)

 

[Gaussian97]

If you are trying to learn mathematics a debate like that is a waste of time.

Well, I think that a very important thing in mathematics is to have very clear what your definitions are, and I see a lot of people doing error because they don't use the definitions properly, so this kind of questions may not be 100% waste of time, because to answer it you need to revisit what exactly is the definition of a trigonometric function, and using the definition search for a consistent argument of why this function may or may not be called trigonometric.

 

[PeroK]

Well, I think that a very important thing in mathematics is to have very clear what your definitions are, and I see a lot of people doing error because they don't use the definitions properly, so this kind of questions may not be 100% waste of time, because to answer it you need to revisit what exactly is the definition of a trigonometric function, and using the definition search for a consistent argument of why this function may or may not be called trigonometric.

Let's say, for the sake of argument:

Person A says that $2\sin x$ is a "trigonometric" function;
Person B says that $2\sin x$ is not a "trigonometric" function; and,
Person C says that there is no formal mathematical definition of a "trigonometric" function. It's just an informal term to group a collection of related functions.

This make no difference to anything.

 

[Gaussian97]

Let's say, for the sake of argument:

Person A says that $2\sin x$ is a "trigonometric" function;
Person B says that $2\sin x$ is not a "trigonometric" function; and,
Person C says that there is no formal mathematical definition of a "trigonometric" function. It's just an informal term to group a collection of related functions.

This make no difference to anything.

Then persons A, B and C try to find an exact definition to 'trigonometric function' where all three agree and using this definition they try to prove if the function fulfils the definition or not.

They have learned what is a mathematical definition, the problems of using concepts with no clear or ambiguous definitions, and also how to prove that a given example fulfils or not a given, perhaps more abstract, definition.

I think that properly done, it's a good academic exercise for someone starting to learn maths.

 

[PeroK]

Then persons A, B and C try to find an exact definition to 'trigonometric function' where all three agree and using this definition they try to prove if the function fulfils the definition or not.

They will never agree because there is no formal definition of a "trigonometric" function. They will end up arguing about dictionary definitions and informal mathematical usage. You may as well argue about whether $\exp(x^2)$ is a "common" function.

 

[Gaussian97]

Person B says that "trigonometric functions" are those function that are elements of the set $\{\sin, \cos, \tan, \csc, \sec, \cot\}$, therefore $2\sin$ is not a trigonometric function.

Person A says that "trigonometric functions" are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.
Therefore, because $2\sin{\theta}=2\frac{opposite}{hypothenuse}$ it is a trigonometric function.

Then Person B asks what exactly means to "relate" etc...

Of course, they can try to agree in one definition and then find whether $2\sin$ is a trigonometric function or not.
Or they may not agree in any definition in which case simply they give the same name to different ideas, which is the most probable, but in any case, it's still a good exercise.

 

[jbriggs444]

but in any case, it's still a good exercise.

It is a pointless exercise since no "important" mathematical fact hinges on the definition and no "important" mathematical theorem is rendered more compact by use of the definition.

By contrast, one can have a reasoned debate on the definition of "prime" because there is a good deal of mathematics which can be stated more compactly and made more general with a particular definition of "prime".

 

[PeroK]

It is a pointless exercise since no "important" mathematical fact hinges on the definition and no "important" mathematical theorem is rendered more compact by use of the definition.

Yes, that is the key point. The mathematical properties of the functions $\sin x$ and $2 \sin x$ are essentially the same.

 

[Gaussian97]

It is a pointless exercise since no "important" mathematical fact hinges on the definition and no "important" mathematical theorem is rendered more compact by use of the definition.

By contrast, one can have a reasoned debate on the definition of "prime" because there is a good deal of mathematics which can be stated more compactly and made more general with a particular definition of "prime".

I think you don't get my point, I'm not saying is something useful or meaningful for you, or for the development of mathematics. It's useful for the person that is learning mathematics, to understand some concepts. In the same way, if I ask an algebra student to multiply two given matrices, it's an academic exercise, but that will lead to some understanding that later on will be useful in some other form.

 

[jbriggs444]

I think you don't get my point, I'm not saying is something useful or meaningful for you, or for the development of mathematics. It's useful for the person that is learning mathematics, to understand some concepts. In the same way, if I ask an algebra student to multiply two given matrices, it's an academic exercise, but that will lead to some understanding that later on will be useful in some other form.

If you are trying to get at a concept like "the 'trigonometric' functions are not closed under multiplication, division, addition, subtraction or function composition", I could see that as being a useful toy exercise.

Though you could stick to more conventional definitions and prove that the irrationals are not closed under multiplication more easily.

 

[PeroK]

To take a concrete example. Discussing whether for integration $x = 2\sin u$ is a "trig" substitution or not would be a waste of time.

 

[FactChecker]

I interpret the term "trigonometric function" as only the specific list of functions, sine, cosine, tangent, cotangent, etc., etc. Nothing else and nothing derived from those.
I do not know if there is an official definition, but I don't remember ever seeing it applied to something like $2\sin(x)$.

 

[vela]

Let's say, for the sake of argument:

Person A says that $2\sin x$ is a "trigonometric" function;
Person B says that $2\sin x$ is not a "trigonometric" function; and,
Person C says that there is no formal mathematical definition of a "trigonometric" function. It's just an informal term to group a collection of related functions.

This make no difference to anything.

It does if they are trying to communicate with others. They need to know the common meaning of the phrase "trigonometric function" lest they confuse themselves or others.

 

[PeroK]

It does if they are trying to communicate with others. They need to know the common meaning of the phrase "trigonometric function" lest they confuse themselves or others.

The evidence of this thread is that we all have our own thoughts on what is and what is not a "trigonometric" function.

 

[hilbert2]

Both $f(x) = \sin x$ and $f(x) = 2\sin x$ (and also $f(x) = \cos x$) are solutions of the same differential equation $f''(x) =-f(x)$, which can be seen as the defining property of the sine and cosine functions. But the tangent function is not a solution of that DE despite being listed as one of the trigonometric functions.

 

[BvU]

Can you tell me why that mistake happens and why you supported it?

arcsin has a domain [-1,1] and a range [-pi/2,pi/2] , sin has a domain $[-\infty,\infty]$ (sorry this ipad is useless) so the plot of arcsin(sin) is a sawtooth

 

[kuruman]

My own thinking is that if $\sin(x)$ is a trigonometric function, $2\sin(x)$ is two trigonometric functions.

 

[archaic]

Would you call $\sin x+\sin x$ trigonometric?

 

[BvU]

Yes. What's the catch this time ?

 

[archaic]

Yes. What's the catch this time ?

It was for the op. Since his function and that one are the same, if he calls the sum trigonometric, then he'd have to call $2\sin x$ trigonometric.

 

[BvU]

The op had a logic statement: "if ... then ... ".

There never was specific oppositon against calling $2\sin x$ trigonometric.

 

[256bits]

Summary:: trigonometric function, composite function

Generally, in encyclopedia, it defines trigonometric function for only 6 types, sin, cos, tan, csc, sec, cot.
Then, here is total conclusion :

those are the ones you are familiar with.
A little bit of history.

Actually there is more, but rarely used today, or rather should say taught, since the rise of computers easing calculations. But when lookup tables were necessary...

https://en.wikipedia.org/wiki/Versine#hav

Whilst the usage of the versine, coversine and haversine as well as their inverse functions can be traced back centuries, the names for the other five cofunctions appear to be of much younger origin.

cofunctions are the ones starting with "co" as in complinentary angle.

Just thought you might want to know that some sources are limited in their scope.

 

[robphy]

For these forgotten trig functions...
http://jwilson.coe.uga.edu/EMAT6680Fa2013/Lively/Forgotten Trig/The_Forgotten_Trigonometric_Functions.pdf
https://www.theonion.com/nation-s-math-teachers-introduce-27-new-trig-functions-1819575558
https://blogs.scientificamerican.co...unctions-your-math-teachers-never-taught-you/(It turns out some of these have applications in special relativity.)