Laplace Transform of sin(3t-2): A Detailed Guide

In this article, we will explore how to find the Laplace transform of the function sin(3t-2). The Laplace transform is a powerful tool used to solve differential equations, and understanding how to apply it to trigonometric functions is crucial for many engineering and mathematical applications.

Understanding the Problem

Given the function sin(3t-2), we aim to find its Laplace transform. The Laplace transform of a function f(t) is defined as:

[ mathcal{L}left{ f(t) right} F(s) int_{0}^{infty} e^{-st} f(t) , dt ]

In this case, ( f(t) sin(3t-2) ). To solve this, we will use the properties of the Laplace transform and trigonometric identities.

Using Trigonometric Identities

We will start by using the sine addition formula to rewrite the function:

[ sin(3t - 2) sin(3t) cos(2) - cos(3t) sin(2) ]

This formula allows us to break down the problem into simpler parts which are easier to handle.

Step-by-Step Solution

Step 1: Laplace Transform of sin(3t)

Using the standard Laplace transform formula for ( sin(at) ), we have:

[ mathcal{L}left{ sin(3t) right} frac{3}{s^2 9} ]

Here, ( a 3 ).

Step 2: Laplace Transform of cos(3t)

The Laplace transform of ( cos(at) ) is given by:

[ mathcal{L}left{ cos(3t) right} frac{s}{s^2 9} ]

Again, ( a 3 ).

Step 3: Combining the Results

Now we can combine these results to find the Laplace transform of the original function:

[ mathcal{L}left{ sin(3t - 2) right} cos(2) cdot left( frac{3}{s^2 9} right) - sin(2) cdot left( frac{s}{s^2 9} right) ]

Simplifying this expression gives us:

[ mathcal{L}left{ sin(3t - 2) right} frac{3cos(2) - ssin(2)}{s^2 9} ]

Verification and Further Exploration

To verify the solution, we can use mathematical software tools like Mathematica or Maple. Running the following commands in Mathematica:

LaplaceTransform[Sin[3t - 2], t, s]

and in Maple with the command:

with(inttrans):
laplace(sin(3t-2), t, s)

Both will return the same result, confirming our manual calculation.

Conclusion

In this guide, we have explored the Laplace transform of ( sin(3t-2) ) by using trigonometric identities and standard Laplace transform properties. Understanding these methods is essential for solving various problems in engineering, physics, and signal processing. The solution can be generalized to other similar functions, making the process adaptable to a wide range of applications.