To find the Laplace transform of $f(t) = \cos^3(t)$, we can break it down using trigonometric identities and then apply the Laplace transform to each component.

Step 1: Use the trigonometric identity for $\cos^3(t)$

We can express $\cos^3(t)$ using a known trigonometric identity: $\cos^3(t) = \frac{1}{4}(3\cos(t) + \cos(3t))$

Step 2: Apply the Laplace transform to each term

Now we take the Laplace transform of each term:

$\mathcal{L}\{ \cos(t) \} = \frac{s}{s^2 + 1}$ $\mathcal{L}\{ \cos(3t) \} = \frac{s}{s^2 + 9}$

Step 3: Combine the results

Now, using the identity we found: $\mathcal{L}\{ \cos^3(t) \} = \frac{1}{4} \left( 3 \cdot \frac{s}{s^2 + 1} + \frac{s}{s^2 + 9} \right)$ Simplifying the expression: $\mathcal{L}\{ \cos^3(t) \} = \frac{s}{4} \left( \frac{3}{s^2 + 1} + \frac{1}{s^2 + 9} \right)$

This is the Laplace transform of $\cos^3(t)$.

Would you like more details on any step? Here are some related questions you might consider:

1. What are the general formulas for Laplace transforms of trigonometric functions?
2. How can we use trigonometric identities to simplify complex expressions?
3. What is the inverse Laplace transform, and how do we use it?
4. How does the Laplace transform help in solving differential equations?
5. Can you find the Laplace transform of other powers of cosine or sine?

Tip: Breaking down a function using known identities often simplifies the process of finding transforms or integrals.