Transforming Trigonometric Functions using Laplace Transformations

[kasse]

I'm trying to transform some functions. These two I haven't succeeded transforming:

f(t) = cos((omega)t + tetha)

f(t) = sint*cost

Also, I need help to find the inverse transform of this function:

F(s) = 8 / (s^2 + 4s)

 

[Dick]

Show us what you tried. It looks to me like you can just take the definition of the laplace transform and integrate by parts twice for the first one.

 

[eugenius]

Hey Dick, I have the same question, laplace of cos(t). I did the integration by parts twice and got to the point where I need to take the limit. Now that's where I'm stuck. There is no limit of cosine and sine. They diverge to infinity. Rather they are bounded between 1 and -1. So what do I do?

Here is my solution.

http://i67.photobucket.com/albums/h304/john_ukranian/48maths.jpg

Too large to paste as an image. Its long but very very clearly defined what I did.

Thanks for the help in advance.

Edit: You can go ahead and write on my image to show mistakes. I noticed that I forgot about the 1/s when subtracting the integral from both sides. That should make a difference, but it still doesn't help with the limit as b goes to infinity.

I know I made mistakes, what are they exactly though.

 

[Dick]

The limits are not a problem sin(t) and cos(t) may oscillate but lim e^(-st) -> 0 at infinity. So evaluated between 0 and infinity sin(t)e^(-st) gives 0 and cos(t)e^(-st) gives -1. And the laplace transform of cos(t) is s/(1+s^2). That's what you are looking for. If A=laplace transform of cos(t), then the integration by parts should lead you to the conclusion A=1/s-A/s^2. Now solve for A.