The Laplace Transform of sin3t: A Comprehensive Guide

Understanding the Laplace transform of sin3t involves the application of trigonometric identities and a series of algebraic manipulations. This article explains the process step-by-step, making it easier to compute the Laplace transform of this specific function.

Introduction to the Laplace Transform

The Laplace transform of a function f(t) is defined as:

?mathcal{L}{f(t)} int_0^∞ e^{-st} f(t) dt

This transform is particularly useful in solving differential equations and analyzing linear time-invariant systems in engineering and science.

Computing the Laplace Transform of sin3t

Directly computing the Laplace transform of sin^3t can be quite complex, so it is often easier to use trigonometric identities to simplify the function before performing the transformation.

Using Trigonometric Identities

The identity for sin^3t is:

sin^3t frac{3sint - sin3t}{4}

Using this identity, we can rewrite sin^3t in a more manageable form.

Step-by-Step Calculation

Laplace Transform of sint is: Laplace Transform of s t ...ight) right frac{1}{4} left 3 cdot frac{1}{s^2 1} - frac{3}{s^2 9} right Combining these results, we get: mathcal{L}{sin^3t} frac{1}{4} left 3 cdot frac{1}{s^2 1} - frac{3}{s^2 9} right Substituting the Laplace transforms, we have: mathcal{L}{sin^3t} frac{1}{4} left 3 cdot frac{1}{s^2 1} - frac{3}{s^2 9} right Now, simplifying the expression: mathcal{L}{sin^3t} frac{1}{4} left frac{3}{s^2 1} - frac{3}{s^2 9} right frac{3}{4} left frac{1}{s^2 1} - frac{1}{s^2 9} right To combine the fractions, we find a common denominator: frac{3}{4} cdot left frac{s^2 9 - s^2 1}{s^2 1s^2 9} right frac{3}{4} cdot frac{8}{s^2 1s^2 9} frac{6}{s^2 1s^2 9}

Thus, the Laplace transform of sin^3t is:

mathcal{L}{sin^3t} frac{6}{s^2 1s^2 9}

Alternative Method Using sin 3t Identity

We can also use the identity:

sin 3t 3 sin t - 4 sin^3 t

Rewriting this, we get:

4sin^3 t 3sin t - sin 3t

Therefore:

sin^3 t frac{3}{4}sin t - frac{1}{4} sin 3t

Applying the Laplace transform to this, we get:

mathcal{L} sin^3 t frac{3}{4}mathcal{L} sin t - frac{1}{4} mathcal{L} sin 3t frac{3}{4 (s^2 1)} - frac{1}{4} frac{3}{s^2 9} frac{3}{4(s^2 1)} - frac{3}{4(s^2 9)}

Further simplifying, we have:

frac{24}{4(s^4 10s^2 9)} frac{6}{s^2 10s^2 9}

Therefore, the Laplace transform of sin^3t:

mathcal{L} { sin^3 t } frac{6}{s^2 10s^2 9}

Conclusion

By leveraging trigonometric identities and algebraic manipulations, we can efficiently compute the Laplace transform of sin^3t. This article provides a comprehensive guide, demonstrating the steps involved in the process and ensuring a proper understanding of the concepts involved.