find laplace transformation of given function 
f(t)=cos^3t

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.

find laplace transformation of given function f(t)=cos^3t

Discover how to find the Laplace transformation of cos^3(t) by applying trigonometric identities and Laplace transform formulas. A step-by-step solution includes breaking down the function using known identities and applying relevant transforms, perfect for college-level or advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation