Laplace Transform of Trigonometric Functions: sin4t and cos4t

The Laplace transform is a fundamental tool in mathematics used to solve differential equations and analyze linear time-invariant systems. It is particularly useful in electrical engineering and control theory. In this article, we focus on calculating the Laplace transform of trigonometric functions, specifically sin4t and cos4t. We will explore the methods and detailed steps to find these transforms.

Introduction to Laplace Transform

The Laplace transform of a function f(t), denoted by mathcal{L}{ft}, is defined as:

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

Laplace Transform of sin4t

Let's start with finding the Laplace transform of sin4t. We can use trigonometric identities to simplify the expression:

sin4t left frac{1 - cos2t}{2} right^2

Expanding this gives:

sin4t frac{1}{4} 1 - 2cos2t cos22t

Next, we use the identity for cos22t:

cos22t frac{1 cos4t}{2}

Substituting this back in, we have:

sin4t frac{1}{4} left 1 - 2cos2t frac{1 cos4t}{2} right

This simplifies to:

sin4t frac{1}{4} left 1 - 2cos2t frac{1}{2} frac{1}{2}cos4t right

Combining terms, we get:

sin4t frac{3}{8} - frac{1}{2}cos2t frac{1}{8}cos4t

Now we can find the Laplace transform of each term separately. The Laplace transform of the individual components are:

mathcal{L}left{frac{3}{8}right} frac{3}{8} cdot frac{1}{s} frac{3}{8s} mathcal{L}left{-frac{1}{2}cos2t right} -frac{1}{2} cdot frac{s}{s^2 4} -frac{s}{2s^2 4} mathcal{L}left{frac{1}{8}cos4t right} frac{1}{8} cdot frac{s}{s^2 16} frac{s}{8s^2 16}

Combining these results, we get:

mathcal{L}{sin^4t} frac{3}{8s} - frac{s}{2s^2 4} frac{s}{8s^2 16}

Laplace Transform of cos4t

Let's now focus on the Laplace transform of cos4t. We can use a similar approach:

cos44t [cos24t]2 left dfrac{1 cos8t}{2} right^2 dfrac{1}{4} 1 cos8t^2

dfrac{1}{4} 1 cos^28t 2cos8t

dfrac{1}{4} left 1 dfrac{1 cos16t}{2} 2cos8t right

dfrac{1}{8} 2 1 cos16t 4cos8t

dfrac{1}{8} 3 cos16t 4cos8t

mathscr{L} cos^4 4t dfrac{1}{8}mathscr{L} 3 cos16t 4cos8t

dfrac{1}{8}left dfrac{3}{s} dfrac{s}{s^2 256} dfrac{4s}{s^2 64} right

Combining the terms, we get:

mathscr{L} cos^4 4t dfrac{1}{8} left[dfrac{3s^2 256s^2 6144}{s^5 - 320s^3 16384s} right]

dfrac{s^4 256s^2 6144}{s^5 - 320s^3 16384s}

Conclusion

In summary, we have successfully calculated the Laplace transforms of both sin4t and cos4t using trigonometric identities and the properties of the Laplace transform. These transforms are essential in various applications in science and engineering, particularly in the analysis and solution of linear systems and differential equations.