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

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In summary: 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.
  • #1
Psi137
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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?
 
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  • #2
Hi ##\psi , \quad## :welcome: !

Psi137 said:
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'.
 
  • #3
Psi137 said:
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.
 
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  • #4
PeroK said:
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
 
  • #5
Psi137 said:
I need the answer of this junk

BvU said:
They are not.
Confirm this and you are in business
 
  • #6
BvU said:
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.
 
  • #8
PeroK said:
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.
 
  • #9
Gaussian97 said:
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.
 
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  • #10
PeroK said:
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.
 
  • #11
Gaussian97 said:
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.
 
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  • #12
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.
 
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  • #13
Gaussian97 said:
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".
 
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  • #14
jbriggs444 said:
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.
 
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  • #15
jbriggs444 said:
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.
 
  • #16
Gaussian97 said:
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.
 
  • #17
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.
 
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  • #18
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)##.
 
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  • #19
PeroK said:
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.
 
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  • #20
vela said:
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.
 
  • #21
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.
 
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  • #22
Psi137 said:
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
 
  • #23
My own thinking is that if ##\sin(x)## is a trigonometric function, ##2\sin(x)## is two trigonometric functions. :oldsmile:
 
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  • #24
Would you call ##\sin x+\sin x## trigonometric?
 
  • #25
Yes. What's the catch this time ?
 
  • #26
BvU said:
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.
 
  • #27
The op had a logic statement: "if ... then ... ".

There never was specific oppositon against calling ##2\sin x## trigonometric.
 
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  • #28
Psi137 said:
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...
1587233356163.png


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.
 
  • #29

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

1. What is a trigonometric function?

A trigonometric function is a mathematical function that relates the angles of a triangle to the lengths of its sides. These functions include sine, cosine, tangent, secant, cosecant, and cotangent.

2. Is 2(sinx) considered a trigonometric function?

Yes, 2(sinx) is considered a trigonometric function because it involves the sine function, which is one of the six basic trigonometric functions.

3. How is 2(sinx) different from other trigonometric functions?

2(sinx) is not significantly different from other trigonometric functions. It is simply a multiple of the sine function, which means it has the same basic properties and behaviors as sine.

4. Can 2(sinx) be simplified?

Yes, 2(sinx) can be simplified to 2sinx. However, this does not change the fact that it is still a trigonometric function.

5. What is the purpose of using trigonometric functions?

Trigonometric functions are used to model and solve problems involving angles and triangles. They have many practical applications in fields such as physics, engineering, and navigation.

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