Stuck on Laplace Transform of Odd Trig Function

[Underhill]

Hey guys!

I'm stuck on a Laplace transform. Following is the problematic function:

[cos(t)]^3

Seems simple, but I'm having issues doing the Laplace transform on odd trigonometric functions. When I use the half-angle formula, I get this, which I can't seem to solve:

1/2cos(t) + 1/2cos(t)*cos(2t)

How do I get this into a form on which I can perform a Laplace transform?

 

[SteamKing]

You can use the cosine addition formula with A = 2t and B = t to obtain an identity which involves (cos t)^3

 

[HallsofIvy]

[tex]cos(3t)= cos(2t)cos(t)- sin(2t)sin(t)= (cos^2(t)- sin^2(t))cos(t)- (2sin(t)cos(t))sin(t)[/tex]
[tex]= cos^3(t)- sin^2(t)cos(t)- 2sin^2(t)cos(t)= cos^3(t)- 3(1- cos^2(t))cos(t)[/tex]
[tex]= 4cos^3(t)- 3cos(t)[/tex]

So [itex]cos^3(t)= cos(3t)/4+ 3cos(t)/4[/itex].

 

[Underhill]

Thanks both of you for your help.

I eventually solved it by using a half-angle formula on [cos(t)]^2, and then using a trig product formula on the resulting expression. I got the same answer as HallsofIvy.

Thanks again, guys!

 

[blue_raver22]

Hi there,

I understand your frustration with the Laplace transform of odd trigonometric functions. It can be tricky to solve, but there are a few steps you can take to simplify the function and make it easier to transform.

First, let's rewrite the function using the double-angle formula for cosine:

[cos(t)]^3 = [cos(t)]^2 * cos(t) = (1 - sin^2(t)) * cos(t)

Next, we can use the substitution u = sin(t) to simplify the function even further:

(1 - u^2) * √(1 - u^2)

Now, we can use the Laplace transform of the square root function, which is 1/s^2, to transform the second term:

(1 - u^2) * √(1 - u^2) = (1 - u^2) * 1/s^2 = 1/s^2 - u^2/s^2

Finally, we can use the Laplace transform of the product of two functions to transform the remaining terms:

L{(1 - u^2) * cos(t)} = L{1/s^2} - L{u^2/s^2} = 1/s^2 - 1/4s^2 = 3/4s^2

Therefore, the Laplace transform of [cos(t)]^3 is:

L{[cos(t)]^3} = 3/4s^2

I hope this helps you solve your Laplace transform. Remember to always simplify the function as much as possible before applying the transform and to use the appropriate Laplace transform rules for each term.

Best of luck with your studies!