Find the Laplace transformation of given function  f(t)=cos^3t

To find the Laplace transform of the given function \( f(t) = \cos^3(t) \), we will first use a trigonometric identity to express \( \cos^3(t) \) in terms of simpler trigonometric functions.

### Step 1: Rewrite \( \cos^3(t) \) using a trigonometric identity.

We use the identity:

\[
\cos^3(t) = \frac{1}{4} \left( 3\cos(t) + \cos(3t) \right)
\]

This simplifies the given function as:

\[
f(t) = \frac{1}{4} \left( 3\cos(t) + \cos(3t) \right)
\]

### Step 2: Apply the Laplace transform to the rewritten function.

Now, we apply the Laplace transform to each term separately.

1. **Laplace transform of \( \cos(t) \):**
   \[
   \mathcal{L}\{ \cos(t) \} = \frac{s}{s^2 + 1}
   \]

2. **Laplace transform of \( \cos(3t) \):**
   \[
   \mathcal{L}\{ \cos(3t) \} = \frac{s}{s^2 + 9}
   \]

### Step 3: Combine the results.

Now, we can combine these results:

\[
\mathcal{L}\{ f(t) \} = \frac{1}{4} \left( 3 \cdot \frac{s}{s^2 + 1} + \frac{s}{s^2 + 9} \right)
\]

Simplifying this expression:

\[
\mathcal{L}\{ f(t) \} = \frac{1}{4} \left( \frac{3s}{s^2 + 1} + \frac{s}{s^2 + 9} \right)
\]

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

Would you like further details or have any questions?

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Here are 5 related questions to expand your understanding:
1. How would the Laplace transform change for \( f(t) = \sin^3(t) \)?
2. What is the inverse Laplace transform of the expression we derived?
3. How can we use Laplace transforms to solve differential equations involving trigonometric functions?
4. What is the Laplace transform of \( \cos(nt) \) for a general \( n \)?
5. How would you solve for the Laplace transform of a product of trigonometric functions?

**Tip:** When dealing with powers of trigonometric functions, always check for useful trigonometric identities to simplify the problem before applying transformations.

Find the Laplace transformation of given function f(t)=cos^3t

Explore the detailed solution for finding the Laplace transform of the function f(t) = cos^3(t). This problem involves using trigonometric identities and applying Laplace transform formulas, ideal for university-level students studying calculus and differential equations.

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