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

  • #1
JJBladester
Gold Member
286
2

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.
 

Answers and Replies

  • #2
135
0
Consider that

[tex]\sin{(4t)}=2\sin{(2t)}\cos{(2t)}[/tex]
 
  • #3
JJBladester
Gold Member
286
2
Consider that

[tex]\sin{(4t)}=2\sin{(2t)}\cos{(2t)}[/tex]
Okay, then [tex]\sin{(2t)}\cos{(2t)} = 1/2\sin{(t)}\cos{(t)}[/tex]

The Laplace integral is then:

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

So where do I go from here to solve this? The [tex]e^{-st}[/tex] is throwing me off, otherwise I'd just do a u substitution with [tex]u = \sin{(t)}[/tex] and [tex]du = \cos{(t)}dt[/tex].
 
  • #4
135
0
Well if you have that

[tex]\sin{(4t)}=2\sin{(2t)}\cos{(2t)}[/tex]

then

[tex]\sin{(2t)}\cos{(2t)}=\frac{1}{2}\sin{(4t)}[/tex]

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

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

where the [itex]s[/itex] you treat as a constant (you're integrating with respect to [itex]t[/itex]). But you could just use the known laplace transform for [itex]\sin{(at)}[/itex] unless you are meaning to derive it.
 
  • #5
JJBladester
Gold Member
286
2
Ahhhh lights coming on. Thanks for the help!
 
  • #6
135
0
Ah good--glad to help.
 

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