Trig Functions Periodicity: Which Function is Not Periodic?

AI Thread Summary
The discussion revolves around identifying which of the given trigonometric functions is not periodic. Participants suggest analyzing the arguments of the functions and their periods, with specific focus on terms like cos(√x) and tan(2x). Graphing is recommended as a method to visualize periodicity, although some participants express discomfort with this approach. Ultimately, it is concluded that sin(√x) is not periodic due to the non-uniform spacing of its values, confirming that it does not repeat at regular intervals. The conversation emphasizes understanding periodicity through both analytical and graphical methods.
  • #51
Pranav-Arora said:
|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?
 
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  • #52
I like Serena said:
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
 
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  • #53
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  • #54
I like Serena said:
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).
 
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  • #55
Pranav-Arora said:
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? :confused:

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 :wink:)?
 
  • #56
I like Serena said:
Yes but why? :confused:

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 :wink:)?

|sin x| period is pi since sin(x) is negative in third and fourth quadrant.:wink: 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
 
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  • #57
Pranav-Arora said:
|sin x| period is pi since sin(x) is negative in third and fourth quadrant.:wink: Applying the modulus to sin(x), the negative values becomes positive and therefore the period become pi

Yep! :smile:

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.
 
  • #58
I like Serena said:
... 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. :confused:

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. :smile:
 
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  • #59
Pranav-Arora said:
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. :smile:

Yes. I think you're getting the hang of it how to interpret graphs! :smile:
Pranav-Arora said:
Sorry i didn't get you. :confused:

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.
 
  • #60
I like Serena said:
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.:smile:

(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) :smile:)[/size]
 
  • #61
PeterO said:
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?
 
  • #62
Bohrok said:
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.
 
  • #63
I like Serena said:
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.
 
  • #64
Pranav-Arora said:
(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) :smile:)[/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. :biggrin:

I can tell you that your trigonometry is still a bit "shaky" as opposed to for instance your logarithms! :smile:
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?
 
  • #65
Bohrok said:
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.
 
  • #66
I like Serena said:
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. :biggrin:

I can tell you that your trigonometry is still a bit "shaky" as opposed to for instance your logarithms! :smile:
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! :smile:
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. :frown:

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. :smile:

Nope :)
we haven't started with matrices.
 
  • #67
Pranav-Arora said:
Thank you for your concern ILS! :smile:
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! :blushing:

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


Pranav-Arora said:
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?
 
  • #68
I like Serena said:
Thanks! :blushing:

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

Your welcome! :smile:

I like Serena said:
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. :wink:
 
  • #69
I like Serena said:
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))
 
  • #70
Bohrok said:
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. :wink:
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.
 
  • #71
I like Serena said:
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)"?

Bohrok said:
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)"?
 
  • #72
Pranav-Arora said:
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.
Pranav-Arora said:
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".
 
  • #73
I like Serena said:
"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?

I like Serena said:
It's math notation for f(g(x)). It's called "function composition" or "f applied to the result of g".
75px-Puzzled.svg.png

Never came across that.
 
  • #74
Pranav-Arora said:
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?
Pranav-Arora said:
75px-Puzzled.svg.png

Never came across that.

You just did! And I expect it will not be the last time. :smile:
 
  • #75
I like Serena said:
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(ex), then it's not periodic since the argument is non linear.
Right..?
 
  • #76
Pranav-Arora said:
If its sin(ex) then the period is \frac{2\pi}{e}. (Found it by applying the sin (nx) rule)
If it is sin(ex), then it's not periodic since the argument is non linear.
Right..?

Right! :smile:

Since I stated it was periodic, it would have to be the first form.
(And anyway, I wouldn't write down an ambiguous expression. :wink:)
 
  • #77
I like Serena said:
Right! :smile:

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


Do you have some more (conceptual)questions for periodicity? :smile:
 
  • #78
Pranav-Arora said:
Do you have some more (conceptual)questions for periodicity? :smile:

No. Don't you have any?
41px-Smiley_green_alien_cool.svg.png
 
  • #79
I like Serena said:
No. Don't you have any?
41px-Smiley_green_alien_cool.svg.png

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

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