Laplace transform of e^(-t/2)cos2(t-1/8π) using translation theorem

[daemon_dkm]

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).

 

[vela]

When you write "cos2(t-1/8π)", which of the following, if any, do you mean?

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

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.

 

[HeisenBerg46]

When you write "cos2(t-1/8π)", which of the following, if any, do you mean?

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

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.