Trig Functions Periodicity: Which Function is Not Periodic?

[Bohrok]

The easiest ones to recognise as non-periodic are those where the argument is an indice.

In sin (x) , sin is the function, x is the argument

sin (x^2) is not periodic [remember x^2 is an indice]

Would it be okay to simply say if the argument is nonlinear, it won't be periodic?

 

[I like Serena]

Would it be okay to simply say if the argument is nonlinear, it won't be periodic?

Nope.
Consider for instance sin(sin x) which is periodic.

 

[Bohrok]

Nope.
Consider for instance sin(sin x) which is periodic.

Touché
I was thinking only of arguments where x was raised to something like a real exponent.
Is it correct to say that the composition of two periodic functions is also periodic? I tried several on WolframAlpha and they were periodic.

 

[I like Serena]

(btw, in a few days (maybe 27th august), a test is going to be conducted in my classes, would you be willing to tell me some short tricks for solving trigonometry questions and other topics. (i will ask other topics to my teacher which are included in the syllabus tomorrow) )[/size]

Hmm, I don't really have a list of tricks ready.
As it is, I have been teaching you tricks during this thread and your previous threads.

I can tell you that your trigonometry is still a bit "shaky" as opposed to for instance your logarithms!
Most important for trigonometry IMHO is understanding and application of the unit circle.
And you would need a list of the trigonometric identities that are not immediately obvious from the unit circle.

There are other topics that I haven't seen any threads of yet (I think).
Like solving equations, or sets of equations.
And like differentiation and integration.
And like sequences, series, and recurrence relations.
Vectors, dot products, and matrices.

Are you supposed to know those as well?

 

[I like Serena]

Touché
I was thinking only of arguments where x was raised to something like a real exponent.
Is it correct to say that the composition of two periodic functions is also periodic? I tried several on WolframAlpha and they were periodic.

What you need is that you can add some constant to x and when you substitute it, you get the same values.
That is you need a constant T, such that for every x you have: f(x) = f(x+T)

As for compositions of periodic functions, try:
sin(x) + 2 sin(e x)
which is not periodic.

 

[Saitama]

Hmm, I don't really have a list of tricks ready.
As it is, I have been teaching you tricks during this thread and your previous threads.

I can tell you that your trigonometry is still a bit "shaky" as opposed to for instance your logarithms!
Most important for trigonometry IMHO is understanding and application of the unit circle.
And you would need a list of the trigonometric identities that are not immediately obvious from the unit circle.

There are other topics that I haven't seen any threads of yet (I think).
Like solving equations, or sets of equations.
And like differentiation and integration.
And like sequences, series, and recurrence relations.
Vectors, dot products, and matrices.

Are you supposed to know those as well?

Thank you for your concern ILS!
Because of especially you and the other members here like SammyS, PeterO, eumyang, Borek (I don't see him on the board now-a-days) i have learned a lot.

I don't understand what do you mean by sets of equation?

Differentiation and integration haven't been really started in my course. Yet we have been given some basic formulas for them. We are done with the chain rule of differentiation. We have been told some basic things like the derivative of displacement is velocity and integrating velocity gives displacement. My teacher said that these things would be taken up next year in much more detail. But i try to learn these using MIT lectures, sadly i get very less time for them.

In my classes, we are done with sequence and series. I try to go back to them because i am sure that i have loads of doubts in it. Sorry but we aren't thought anything like "recurrence relations."

Vectors is really easy for me and my teacher has made it so easy for us that we feel like that's the most easiest topic in physics. I rarely get doubts, and if doubts occur, my physics teacher explains it.

Nope :)
we haven't started with matrices.

 

[I like Serena]

Thank you for your concern ILS!
Because of especially you and the other members here like SammyS, PeterO, eumyang, Borek (I don't see him on the board now-a-days) i have learned a lot.

Thanks!

Last I heard, Borek was on a vacation, but it has indeed been quite a while now.

I don't understand what do you mean by sets of equation?

Oh, that's like:

Suppose the sum of the ages of Amy and Bria is 28, and the product of their ages is 195, what are their respective ages?

 

[Saitama]

Thanks!

Last I heard, Borek was on a vacation, but it has indeed been quite a while now.

Your welcome!

Oh, that's like:

Suppose the sum of the ages of Amy and Bria is 28, and the product of their ages is 195, what are their respective ages?

These type of questions i used to do in past.

 

[Bohrok]

What you need is that you can add some constant to x and when you substitute it, you get the same values.
That is you need a constant T, such that for every x you have: f(x) = f(x+T)

As for compositions of periodic functions, try:
sin(x) + 2 sin(e x)
which is not periodic.

That's a sum of periodic functions; I meant like (f o g)(x), such as sin(sin(x)) and cos(tan(x))

 

[I like Serena]

That's a sum of periodic functions; I meant like (f o g)(x), such as sin(sin(x)) and cos(tan(x))

Oh, all right.
A little sharper is that if each inner function which is taken from x is periodic, the result will be periodic.
Note that if there is more than one function that is taken from x, they all need to be periodic and the ratio of their periods must be a rational number.

 

[Saitama]

What you need is that you can add some constant to x and when you substitute it, you get the same values.
That is you need a constant T, such that for every x you have: f(x) = f(x+T)

As for compositions of periodic functions, try:
sin(x) + 2 sin(e x)
which is not periodic.

What's this "sin(e x)"?

That's a sum of periodic functions; I meant like (f o g)(x), such as sin(sin(x)) and cos(tan(x))

And what's this "(f o g)(x)"?

 

[I like Serena]

What's this "sin(e x)"?

"e" is Euler's number (2.71828), which is the base of the natural logarithm.
I used it because it's an irrational number other than pi.
In particular the ratio between e and pi cannot be written as the ratio of 2 whole numbers.

And what's this "(f o g)(x)"?

It's math notation for f(g(x)). It's called "function composition" or "f applied to the result of g".

 

[Saitama]

"e" is Euler's number (2.71828), which is the base of the natural logarithm.
I used it because it's an irrational number other than pi.
In particular the ratio between e and pi cannot be written as the ratio of 2 whole numbers.

Is x is raised to the power of e in sin(e x) or is x multiplied to e?

It's math notation for f(g(x)). It's called "function composition" or "f applied to the result of g".

Never came across that.

 

[I like Serena]

Is x is raised to the power of e in sin(e x) or is x multiplied to e?

What is the period in each case?

Never came across that.

You just did! And I expect it will not be the last time.

 

[Saitama]

What is the period in each case?

If its sin(ex) then the period is \frac{2\pi}{e}. (Found it by applying the sin (nx) rule)
If it is sin(e^x), then it's not periodic since the argument is non linear.
Right..?

 

[I like Serena]

If its sin(ex) then the period is \frac{2\pi}{e}. (Found it by applying the sin (nx) rule)
If it is sin(e^x), then it's not periodic since the argument is non linear.
Right..?

Right!

Since I stated it was periodic, it would have to be the first form.
(And anyway, I wouldn't write down an ambiguous expression. )

 

[Saitama]

Right!

Since I stated it was periodic, it would have to be the first form.
(And anyway, I wouldn't write down an ambiguous expression. )

Do you have some more (conceptual)questions for periodicity?

 

[I like Serena]

Do you have some more (conceptual)questions for periodicity?

No. Don't you have any?

 

[Saitama]

No. Don't you have any?

No.
(You found out the website from where i am using these emoticons. )