Problem 39 - Advanced Engineering Mathematics

Question

Find \(\mathscr{L}\\{f(t)\\}\) by first using an appropriate trigonometric identity. \(f(t) \quad \sin (4 t+5)\)

Step-by-Step Solution

Answer

\[ \mathscr{L} \{ f(t) \} = \cos(5) \cdot \frac{4}{s^2 + 16} + \sin(5) \cdot \frac{s}{s^2 + 16} \]

Step 1: Identify the Trigonometric Identity

We start by identifying an applicable trigonometric identity. For the function \(f(t) = \sin(4t + 5)\), we use the identity \(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\).

Step 2: Apply the Identity to Expand the Expression

Using the above identity, we expand \(\sin(4t + 5)\): \[ \sin(4t+5) = \sin(4t)\cos(5) + \cos(4t)\sin(5) \] This expression can now be handled in parts for the Laplace transform.

Step 3: Find the Laplace Transform of Each Term

The Laplace transform of a sine function \(\sin(at)\) is \(\mathscr{L}\{\sin(at)\} = \frac{a}{s^2 + a^2}\) and for a cosine function \(\mathscr{L}\{\cos(at)\} = \frac{s}{s^2 + a^2}\). Apply these to the parts: - \(\mathscr{L}\{\sin(4t)\cos(5)\} = \cos(5) \cdot \frac{4}{s^2 + 16}\) - \(\mathscr{L}\{\cos(4t)\sin(5)\} = \sin(5) \cdot \frac{s}{s^2 + 16}\)

Step 4: Combine the Transforms into the Final Result

Add the individual Laplace transforms to get the final result:\[ \mathscr{L} \{ f(t) \} = \cos(5) \cdot \frac{4}{s^2 + 16} + \sin(5) \cdot \frac{s}{s^2 + 16} \]

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that relate the angles and sides of triangles. They help simplify complex trigonometric expressions, making them easier to work with. In our exercise, we used the trigonometric identity for the sine of a sum: \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \). This identity is useful for splitting the function \( \sin(4t + 5) \) into manageable parts. By expanding \( \sin(4t + 5) \), we express it as a sum of sine and cosine functions: \[ \sin(4t+5) = \sin(4t)\cos(5) + \cos(4t)\sin(5) \]. This sets the stage for applying the Laplace transform effectively, using known formulas for sine and cosine functions.

Laplace Transform of Sine and Cosine

The Laplace transform is a powerful tool for transforming a time-domain function into a frequency-domain function. This is particularly helpful in solving differential equations, analyzing systems, and studying signals. For sine and cosine functions like \( \sin(at) \) and \( \cos(at) \), the Laplace transforms are

• \( \mathscr{L}\{\sin(at)\} = \frac{a}{s^2 + a^2} \)
• \( \mathscr{L}\{\cos(at)\} = \frac{s}{s^2 + a^2} \)

These formulas demonstrate how sine and cosine functions are transformed into rational functions of \( s \), where \( s \) is a complex number. By applying these formulas, we can convert exponential oscillating functions into simpler algebraic terms, easing the analysis and problem-solving process.

Analysis of Functions

Analyzing functions involves understanding their behavior and properties. This process is critical in the application of the Laplace transform. First, we recognize that our function \( f(t) = \sin(4t + 5) \) can be broken down using a trigonometric identity. By doing so, we create simpler components that are easier to transform.In this specific case, splitting \( \sin(4t+5) \) into \( \sin(4t)\cos(5) + \cos(4t)\sin(5) \), allows each part to be separately evaluated using the Laplace transform. Breaking functions into smaller parts helps handle each section with known solutions, streamlining the computational procedure. This way, even a complex expression can be systematically dealt with.

Mathematical Problem Solving

Mathematical problem solving often requires the application of principles and techniques such as trigonometric identities and the Laplace transform. Being systematic and methodical is key. Here are steps to approach similar problems:

• Identify applicable identities: Determine relevant mathematical identities or properties that can simplify the function.
• Break down the expression: Use the identified identities to decompose the expression into solvable parts.
• Transform each component: Apply appropriate transform formulas to each part to simplify further.
• Combine solutions: Aggregate all parts' transforms for the final result.

This structured approach guides through understanding and solving complex problems. Remember, practicing these step-by-step methods boosts problem-solving skills, enabling tackling more challenging mathematical exercises efficiently.