To find the Laplace transform of the function \( 2 \cos(3t) \), we use the standard Laplace transform formula for the cosine function:

\[
\mathcal{L}[\cos(at)] = \frac{s}{s^2 + a^2}
\]

Given \( 2 \cos(3t) \), here \( a = 3 \) and we have a coefficient of 2 multiplying the cosine. The Laplace transform of \( 2 \cos(3t) \) can be found by multiplying the Laplace transform of \( \cos(3t) \) by 2:

\[
\mathcal{L}[2 \cos(3t)] = 2 \mathcal{L}[\cos(3t)]
\]

Using the formula:

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

Thus,

\[
\mathcal{L}[2 \cos(3t)] = 2 \cdot \frac{s}{s^2 + 9} = \frac{2s}{s^2 + 9}
\]

So, the Laplace transform of \( 2 \cos(3t) \) is:

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

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. What is the Laplace transform of \( \sin(4t) \)?
2. How does the Laplace transform change for \( e^{2t} \cos(3t) \)?
3. What is the Laplace transform of a general function \( a \cos(bt) + b \sin(at) \)?
4. How can we apply the Laplace transform to solve differential equations involving trigonometric functions?
5. What is the inverse Laplace transform of \( \frac{s}{s^2 + 16} \)?

**Tip:** When computing Laplace transforms of trigonometric functions, it helps to remember the standard forms for sine and cosine, as well as the linearity property, which allows you to multiply by constants.

Explore how to compute the Laplace transform of 2 times the cosine of 3t using the standard formula. This detailed explanation covers the method to find the Laplace transform of cosine functions with specific coefficients. Suitable for advanced mathematics enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation