Laplace transform of cos2(t-1/8π), help?

In summary: So, you need to first use trig identities to get the expression into one of those forms, and then you can use the translation theorem to solve for the Laplace transform. Overall, it seems like you have a good understanding of the method, you just need to figure out how to apply it to the specific function given in the problem.
  • #1
daemon_dkm
6
0

Homework Statement


I'm working on some Differential Equations homework and I'm stuck.

The question is apply the translation theorem to find the Laplace transform.

f(t) = e^(-t/2)cos2(t-1/8π)

I know how to apply the method, I just need to figure out how to transform the cosine function. (I think)

The answer should be: √(2) (2s+5)/(4s²+4s+17)


Homework Equations


L{f(t)*e^at)} = f(s-a)


The Attempt at a Solution



I'm completely stuck on how to transform cos2(t-1/8π).
 
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  • #2
When you write "cos2(t-1/8π)", which of the following, if any, do you mean?

[tex]\begin{align*}
&\cos^2 \left(t-\frac{1}{8\pi}\right) \\
&\cos \left[2\left(t-\frac{1}{8\pi}\right)\right] \\
&\cos^2 \left(t-\frac{\pi}{8}\right) \\
&\cos \left[2\left(t-\frac{\pi}{8}\right)\right]
\end{align*}[/tex]

For any of those cases, try using trig identities to turn it into a form where you can see how to get the Laplace transform.
 
  • #3
vela said:
When you write "cos2(t-1/8π)", which of the following, if any, do you mean?

[tex]\begin{align*}
&\cos^2 \left(t-\frac{1}{8\pi}\right) \\
&\cos \left[2\left(t-\frac{1}{8\pi}\right)\right] \\
&\cos^2 \left(t-\frac{\pi}{8}\right) \\
&\cos \left[2\left(t-\frac{\pi}{8}\right)\right]
\end{align*}[/tex]

For any of those cases, try using trig identities to turn it into a form where you can see how to get the Laplace transform.

The answer really depends on which one you mean. Remember that the laplace transform of cos(a*t)=s/(s^2+a^2) for a real number a and s/(s^2+a) for complex s.
 

1. What is a Laplace transform?

A Laplace transform is a mathematical operation that converts a function of time into a function of frequency. It is commonly used in engineering and physics to solve differential equations and analyze systems.

2. How do you calculate the Laplace transform of cos2(t-1/8π)?

To calculate the Laplace transform of cos2(t-1/8π), you can use the formula: L{cos(at+b)} = s/(s^2+a^2) - (a/s^2+a^2)cos(b). In this case, a=2 and b=-1/8π. So, L{cos2(t-1/8π)} = s/(s^2+4) - (2/s^2+4)cos(-1/8π).

3. What is the significance of the Laplace transform of cos2(t-1/8π)?

The Laplace transform of cos2(t-1/8π) represents the frequency domain representation of the original function. It can be used to analyze the behavior of the function at different frequencies and can help in solving differential equations involving the function.

4. Are there any real-world applications of Laplace transform of cos2(t-1/8π)?

Yes, the Laplace transform of cos2(t-1/8π) has many real-world applications in fields such as electrical engineering, control systems, and signal processing. It is used to model and analyze systems with periodic inputs and can help in designing filters and control systems.

5. Is there a way to visualize the Laplace transform of cos2(t-1/8π)?

Yes, the Laplace transform of cos2(t-1/8π) can be visualized using a graph of the magnitude and phase of the function in the frequency domain. This can help in understanding the behavior of the function at different frequencies and can aid in solving problems related to the function.

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