fundamental period

[renegade05]

Looking for the fundamental period of the following functions. It has been a while since I had to do this and kind of forgot the tricks of the trade.

Thanks!

(a) \(\displaystyle \frac{1}{2+cos(3x)}\)

(b) \(\displaystyle e^{-cos^2x}\)

Ok sure I know for (a) cos(3x) has a period of \(\displaystyle \frac{2\pi}{3}\) but I am not sure how to proceed from there.

and (b) I am really not sure how to incorporate the exponential.

Thanks!

 

[Ishuda]

Looking for the fundamental period of the following functions. It has been a while since I had to do this and kind of forgot the tricks of the trade.

Thanks!

(a) \(\displaystyle \frac{1}{2+cos(3x)}\)

(b) \(\displaystyle e^{-cos^2x}\)

Ok sure I know for (a) cos(3x) has a period of \(\displaystyle \frac{2\pi}{3}\) but I am not sure how to proceed from there.

and (b) I am really not sure how to incorporate the exponential.

Thanks!

For (a), you are very very close. You need to also remember that the fundamental period of a function is the shortest repeat cycle (shortest period). So if f(x+a) = f(x) and a is the minimum value for which this happens for all x (in the domain of f), a is the fundamental period. You know if f(x) = cos(3x), the fundamental period of f is \(\displaystyle \frac{2\pi}{3}\). If f(x) were \(\displaystyle \frac{1}{2+cos(3x)}\) wouldn't f(x+\(\displaystyle \frac{2\pi}{3}\)) = f(x). Would there be any other value less that that which would make that true?

For (b), what is the fundamental period of cos²?

 

[renegade05]

For (a), you are very very close. You need to also remember that the fundamental period of a function is the shortest repeat cycle (shortest period). So if f(x+a) = f(x) and a is the minimum value for which this happens for all x (in the domain of f), a is the fundamental period. You know if f(x) = cos(3x), the fundamental period of f is \(\displaystyle \frac{2\pi}{3}\). If f(x) were \(\displaystyle \frac{1}{2+cos(3x)}\) wouldn't f(x+\(\displaystyle \frac{2\pi}{3}\)) = f(x). Would there be any other value less that that which would make that true?

For (b), what is the fundamental period of cos²?

for (a) no. So the fundamental period is \(\displaystyle \frac{2\pi}{3}\)

for (b) fundamental of \(\displaystyle cos^2(x)\) is \(\displaystyle \pi\) hmmm.. I guess the exponential has no effect on it. so \(\displaystyle \pi\) is the answer.

 

[Deleted member 4993]

Another way would be to look at the Taylor's series expansion of the given functions.