What is the Laplace transform of sin2tcos2t?

In summary, the problem is to find the laplace transform of f(t) = sin2tcos2t and the correct answer is (2/(s^2 + 16)). To solve this, one can use the trig identity sin2x=2sinxcosx and the laplace transform of f(t) = 1/4sin^2(2t). Another approach is to look up the transform in a table and work backwards.
  • #1
RafiS
1
0
iv got a problem i can't seem to understand. if anyone could help me out it would be great

f(t)= sin2tcos2t

im just not sure what to do when i have the product of 2 trig functions.

the correct answer is (2/(s^2 + 16))

thanks for any help
 
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  • #2
Is there something you can do with f(t) to make it look different?
 
  • #3
yeah, like exk suggested, try to use this trig identity somehow:

[tex] sin2x=2isnxcosx[/tex] can you figure it out how to transform your f(t) into a similar form?
 
  • #4
Use f(x) = 1/4sin^2(2t)
f'(x) = sin(2t)cos(2t)
 
  • #5
Vid said:
Use f(x) = 1/4sin^2(2t)
f'(x) = sin(2t)cos(2t)

I don't see how would this help!
If i have gotten the op right, he just needst to take the laplace transform of

[tex]f(t)=sin2tcos2t=\frac{1}{2}sin4t[/tex]
so

[tex]L{f(t)}=L{\frac{1}{2}sin4t}=\int_{0}^{\infty}\frac{1}{2}sin(4t)e^{-st}dt[/tex]
 
  • #6
I was thinking that sin^2(t) was a common laplace transform, but I was mistaken.
 
  • #7
another strategy for this would be to look up in a transform table what your answer corresponds to and work backwards.

sutupidmath's solution is correct.
 

1. What is the Laplace transform?

The Laplace transform is a mathematical tool used to convert a function of time into a function of frequency. It is commonly used in engineering and physics to solve differential equations and analyze dynamic systems.

2. Why do we use the Laplace transform?

The Laplace transform allows us to simplify and solve complex differential equations that may be difficult to solve using other methods. It also helps in understanding the behavior of systems over time and frequency domains.

3. How do you perform a Laplace transform?

To perform a Laplace transform, you need to integrate the given function of time multiplied by the exponential function e^-st, where s is a complex variable. The result is a function of frequency in terms of s.

4. What are the applications of Laplace transform?

The Laplace transform has various applications in engineering, physics, and mathematics. It is used to analyze electrical circuits, control systems, signal processing, and mechanical systems. It is also used in probability and statistics to solve problems involving random variables.

5. Are there any limitations to the Laplace transform?

One limitation of the Laplace transform is that it can only be applied to functions that are defined for all positive values of time. It also assumes that the function is well-behaved and does not have any discontinuities or infinite discontinuities. Additionally, the inverse Laplace transform may not always exist for every function.

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