Laplace transform of sin(2t)cos(2t)

[JJBladester]

Homework Statement

Find the Laplace transform of

f(t) = sin(2t)cos(2t)

using a trig identity.

Homework Equations

N/A.

The Attempt at a Solution

I know the double-angle formula sin(2t) = 2sin(t)cos(t) but that's not helping much. Can you give me some advice on how to proceed.

 

[jeffreydk]

Consider that

\sin{(4t)}=2\sin{(2t)}\cos{(2t)}

 

[JJBladester]

Consider that

\sin{(4t)}=2\sin{(2t)}\cos{(2t)}

Okay, then \sin{(2t)}\cos{(2t)} = 1/2\sin{(t)}\cos{(t)}

The Laplace integral is then:

1/2\int{e^{-st}\sin{(t)}\cos{(t)}}dt

So where do I go from here to solve this? The e^{-st} is throwing me off, otherwise I'd just do a u substitution with u = \sin{(t)} and du = \cos{(t)}dt.

 

[jeffreydk]

Well if you have that

\sin{(4t)}=2\sin{(2t)}\cos{(2t)}

then

\sin{(2t)}\cos{(2t)}=\frac{1}{2}\sin{(4t)}

So then if you want to do it from the integral you just need to integrate

\int_0^{\infty}\frac{1}{2}\sin{(4t)}e^{-st}dt

where the s you treat as a constant (you're integrating with respect to t). But you could just use the known laplace transform for \sin{(at)} unless you are meaning to derive it.

 

[JJBladester]

Ahhhh lights coming on. Thanks for the help!

 

[jeffreydk]

Ah good--glad to help.