Trig Functions Periodicity: Which Function is Not Periodic?

[Saitama]

May i know what's the period of tan²x because i am getting stuck in the (b) and (c) option.

 

[PeterO]

May i know what's the period of tan²x because i am getting stuck in the (b) and (c) option.

Probably pi.

 

[SammyS]

What is an indice?

The plural of index is indices, so by "indice", Peter O probably means index.

However, I think he really is talking about an exponent, that is to say a power of x, not a power of the trig function.

 

[Saitama]

Probably pi.

How? I Like Serena gave me an equation to solve it and deduce the period of sin²x in https://www.physicsforums.com/showpost.php?p=3448043&postcount=16". How would i solve that equation?

The plural of index is indices, so by "indice", Peter O probably means index.

However, I think he really is talking about an exponent, that is to say a power of x, not a power of the trig function.

Thanks for solving the confusion,

 

[I like Serena]

How? I Like Serena gave me an equation to solve it and deduce the period of sin²x in https://www.physicsforums.com/showpost.php?p=3448043&postcount=16". How would i solve that equation?

Try tan²x = sin²x / cos²x and apply the cos 2x formulas.

 

[Saitama]

Try tan²x = sin²x / cos²x and apply the cos 2x formulas.

Applying cos(2x) formulas, i get $$tan^2x=\frac{1-cos(2x)}{cos(2x)+1}$$.
What next?

 

[I like Serena]

Applying cos(2x) formulas, i get $$tan^2x=\frac{1-cos(2x)}{cos(2x)+1}$$.
What next?

What is the period of cos(2x)?

 

[Saitama]

What is the period of cos(2x)?

Is it pi?

 

[I like Serena]

Is it pi?

Perhaps. Why would it be pi?

 

[Saitama]

Perhaps. Why would it be pi?

Since cos(x)=cos(x+2pi)=cos(x+4pi)...
therefore cos(2x)=cos(2x+2pi)=cos(2x+4pi)...
or cos2(x)=cos 2(x+pi)=cos 2(x+2pi)...
So, the period is pi.

Btw, is this relation correct? If a function is given, cos n(x), where n is an integer then its period would be $\frac{2\pi}{n}$.

 

[I like Serena]

Since cos(x)=cos(x+2pi)=cos(x+4pi)...
therefore cos(2x)=cos(2x+2pi)=cos(2x+4pi)...
or cos2(x)=cos 2(x+pi)=cos 2(x+2pi)...
So, the period is pi.

Btw, is this relation correct? If a function is given, cos n(x), where n is an integer then its period would be $\frac{2\pi}{n}$.

Yep!

So the period of tan²x...?

 

[Saitama]

Yep!

So the period of tan²x...?

Is it pi? If it is so, then is it becuase the period of cos(2x) is pi?

 

[I like Serena]

Is it pi? If it is so, then is it becuase the period of cos(2x) is pi?

Yes.
Note that if you evaluate the function for x+pi instead of for x, you'll get the same result.

 

[I like Serena]

Btw, there is a catch.

Consider for instance |sin x| and |1 + sin x|.
What are their respective periods?

 

[Saitama]

Btw, there is a catch.

Consider for instance |sin x| and |1 + sin x|.
What are their respective periods?

|sin x| period is pi. First i thought that the period for |1+sin x| is also pi but when i checked it on wolfram alpha it is 2pi. Why is it so?

 

[I like Serena]

|sin x| period is pi. First i thought that the period for |1+sin x| is also pi but when i checked it on wolfram alpha it is 2pi. Why is it so?

What did you see on WolframAlpha?

 

[Saitama]

What did you see on WolframAlpha?

This is the link:-http://www.wolframalpha.com/input/?i=|1+sin+x|"

[PLAIN]http://www3.wolframalpha.com/Calculate/MSP/MSP153119ggif1081a6cdhb00005f313i3bi0hec616?MSPStoreType=image/gif&s=40&w=185&h=18

 

[I like Serena]

This is the link:-http://www.wolframalpha.com/input/?i=|1+sin+x|"

[PLAIN]http://www3.wolframalpha.com/Calculate/MSP/MSP153119ggif1081a6cdhb00005f313i3bi0hec616?MSPStoreType=image/gif&s=40&w=185&h=18[/QUOTE]

All right, here's my counter:
http://www.wolframalpha.com/input/?i=|sin+x|,+|1+sin+x|"

Can you interpret what you see in the graph?

 

[Saitama]

All right, here's my counter:
http://www.wolframalpha.com/input/?i=|sin+x|,+|1+sin+x|"

Can you interpret what you see in the graph?

I can interpret that |sin(x)| is periodic with pi and |1+sin(x)| is periodic with 2pi. Also both the graphs intersect at two points between (0,2pi).

 

[I like Serena]

I can interpret that |sin(x)| is periodic with pi and |1+sin(x)| is periodic with 2pi. Also both the graphs intersect at two points between (0,2pi).

Yes but why?

Actually, since sin x has a period of 2pi, one might expect that |sin x| also has a period of 2pi.
How come it has a shorter period (other than that Wolfram says so )?

 

[Saitama]

Yes but why?

Actually, since sin x has a period of 2pi, one might expect that |sin x| also has a period of 2pi.
How come it has a shorter period (other than that Wolfram says so )?

|sin x| period is pi since sin(x) is negative in third and fourth quadrant. Applying the modulus to sin(x), the negative values becomes positive and therefore the period become pi and it results in a graph like this:-
[PLAIN]http://www3.wolframalpha.com/Calculate/MSP/MSP46719gh00f8c6ai9f2100003i3ic66d1763h0b2?MSPStoreType=image/gif&s=35&w=299&h=142&cdf=Coordinates&cdf=Tooltips

 

[I like Serena]

|sin x| period is pi since sin(x) is negative in third and fourth quadrant. Applying the modulus to sin(x), the negative values becomes positive and therefore the period become pi

Yep!

More generally, we can say that if you know that a function repeats itself after for instance 2pi (like |sin x|), that it is still possible that the actual (shortest) period of the function is shorter, but it will have to be a divider of the period you found.

 

[Saitama]

... that it is still possible that the actual (shortest) period of the function is shorter, but it will have to be a divider of the period you found.

Sorry i didn't get you.

Btw, i got why the period of |1+sin(x)| is 2pi. The graph of sin(x) is[PLAIN]http://www3.wolframalpha.com/Calculate/MSP/MSP133419ggii6778e7h11g0000594571daf619f270?MSPStoreType=image/gif&s=2&w=299&h=131&cdf=Coordinates&cdf=Tooltips

If we add one to sin(x), that means we are adding one to all the outputs of sin(x) which makes the graph to flow over zero. Now that means if we apply the modulus function, it doesn't affect the graph since all the values of 1+sin(x) are positive. Therefore the period of |1+sin(x)| is 2pi.

 

[I like Serena]

Btw, i got why the period of |1+sin(x)| is 2pi.

If we add one to sin(x), that means we are adding one to all the outputs of sin(x) which makes the graph to flow over zero. Now that means if we apply the modulus function, it doesn't affect the graph since all the values of 1+sin(x) are positive. Therefore the period of |1+sin(x)| is 2pi.

Yes. I think you're getting the hang of it how to interpret graphs!

Sorry i didn't get you.

I just meant, that as far as you can tell without looking at the graph, at first you would assume that |sin x| has period 2pi.
When you look at the graph, or if you otherwise think about it some more, you'd see that in this case the actual (shortest) period is pi, which is half of 2pi.

This is also why it is so important to look at the graph and interpret it.

 

[Saitama]

I just meant, that as far as you can tell without looking at the graph, at first you would assume that |sin x| has period 2pi.
When you look at the graph, or if you otherwise think about it some more, you'd see that in this case the actual (shortest) period is pi, which is half of 2pi.

This is also why it is so important to look at the graph and interpret it.

Yeah, at first i thought that the period of |sin(x)| is 2pi but my teacher corrected me that i didn't take care of modulus function which is the absolute value function.

(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) )

 

[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) )

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.