Find which functions have the same period

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Loki123

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I thought f3 and f5 also have the same period but I was wrong. Why? What's the period for sinx^2 and, out of curiosity, what'the period for sin^2(x)
 
What is f3 and f5?

Look at the graphs of sinx^2 and sin^2x and see if there is a period. If there is, then what is it? Then prove that this result is correct.
 
Jomo said:
What is f3 and f5?

Look at the graphs of sinx^2 and sin^2x and see if there is a period. If there is, then what is it? Then prove that this result is correct.
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I would prefer to do this without a graph if that's possible.
 
Use double angle formula [math]\cos(2x)=1-2\sin^2(x) \Rightarrow \sin^2(x) =\frac{1}{2}(1-\cos(2x))[/math]What's the period of the last expression?
[imath]\sin(x^2)[/imath] is not periodic
 
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Loki123 said:
I would prefer to do this without a graph if that's possible.
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I don't know how to find the period of a function in general. If you think the period is P, then you can confirm it by proving that f(x+P) = f(x).
 
Loki123 said:
I would prefer to do this without a graph if that's possible.
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I respect that, but still look at the graph just to confirm that it is periodic. If it is not periodic then try to show why that is so and if it is periodic then show that it is. This is the way to do mathematics. Use the graph just to confirm whether if it is or is not periodic.
 
BigBeachBananas said:
Use double angle formula [math]\cos(2x)=1-2\sin^2(x) \Rightarrow \sin^2(x) =\frac{1}{2}(1-\cos(2x))[/math]What's the period of the last expression?
[imath]\sin(x^2)[/imath] is not periodic
Click to expand...
Pi is the period of cos2x, is that the period of sin^2x?
 
BigBeachBananas said:
Use double angle formula [math]\cos(2x)=1-2\sin^2(x) \Rightarrow \sin^2(x) =\frac{1}{2}(1-\cos(2x))[/math]What's the period of the last expression?
[imath]\sin(x^2)[/imath] is not periodic
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Why is sinx^2 not periodic? Does that also apply to cosx^2, tanx^2 and cotx^2?
 
Loki123 said:
Pi is the period of cos2x
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Correct. Multiply or adding constant does not change the period of [imath]\cos(2x)[/imath]. You have the general transformation of trigonometric function in the OP. The period is determined by [imath]\omega = \frac{2\pi}{|B|}[/imath]
Loki123 said:
Why is sinx^2 not periodic? Does that also apply to cosx^2, tanx^2 and cotx^2?
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You can prove it by using what @lev888 said in post#6.
For a function f(x) to be periodic with a period P, we must have

[math]f(x+P)=f(x) ∀x[/math]This implies that [imath]f(x)=f(x+P)=f(x+2P)=f(x+3P)=\dots[/imath] must be true. So, a periodic function of x repeats after equally spaced values of P.

Now, for [imath]\sin(x^2)[/imath]. Suppose [imath]x^2 =0, \pi, 2\pi,3\pi\dots[/imath]. Although these values are equally spaced, the
corresponding values [imath]x = 0, \sqrt{π}, \sqrt{2π},\sqrt{3π},\dots[/imath] are not equally spaced.

Use a similar idea for the other trig functions.
 
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BigBeachBananas said:
Correct. Multiply or adding constant does not change the period of [imath]\cos(2x)[/imath]. You have the general transformation of trigonometric function in the OP. The period is determined by [imath]\omega = \frac{2\pi}{|B|}[/imath]

You can prove it by using what @lev888 said in post#6.
For a function f(x) to be periodic with a period P, we must have

[math]f(x+P)=f(x) ∀x[/math]This implies that [imath]f(x)=f(x+P)=f(x+2P)=f(x+3P)=\dots[/imath] must be true. So, a periodic function of x repeats after equally spaced values of P.

Now, for [imath]\sin(x^2)[/imath]. Suppose [imath]x^2 =0, \pi, 2\pi,3\pi\dots[/imath]. Although these values are equally spaced, the
corresponding values [imath]x = 0, \sqrt{π}, \sqrt{2π},\sqrt{3π},\dots[/imath] are not equally spaced.

Use a similar idea for the other trig functions.
Click to expand...
Well it would be the same for all of them since they would have have square roots. So trigx^2 is not periodic.

How about trig^2(x)
We know for sin^2(x) it's pi.
Is forr cos^2(x) pi?
For tan^2(x) and cot^2(x) I am having trouble.
 
Loki123 said:
Well it would be the same for all of them since they would have have square roots. So trigx^2 is not periodic.

How about trig^2(x)
We know for sin^2(x) it's pi.
Is forr cos^2(x) pi?

For tan^2(x) and cot^2(x) I am having trouble.
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The half-angle identities can be re-stated by squaring each side and doubling all of the angle measures
1641591310889.png
 
Loki123 said:
Yes i got that, but how do I determine period if cos2x is diving cos2x?
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I believe you're referring to [imath]\tan^2(x)[/imath]. It's been a while but it can be shown that the periodicity of [imath]\tan^n(x)= \pi[/imath]
 
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