Problem 38
Question
Find \(\mathscr{L}\\{f(t)\\}\) by first using an appropriate trigonometric identity. \(f(t) \quad \cos ^{2} t\)
Step-by-Step Solution
Verified Answer
\( \mathscr{L}\{f(t)\} = \frac{1}{2s} + \frac{s}{2(s^2 + 4)} \).
1Step 1: Identify the Trigonometric Identity
To solve this problem, first identify a trigonometric identity that can simplify \( \cos^2 t \). The relevant identity is: \[ \cos^2 t = \frac{1 + \cos(2t)}{2}. \] Use this identity to express \( f(t) = \cos^2 t \) in a simpler form.
2Step 2: Apply the Identity to the Function
Substitute the identity into the original function: \[ f(t) = \frac{1 + \cos(2t)}{2}. \] This step simplifies the function into terms of constants and a cosine function, which are easier to work with in Laplace transformations.
3Step 3: Break Down the Function for Laplace Transform
Write the function as a sum of two terms: \[ f(t) = \frac{1}{2} + \frac{\cos(2t)}{2}. \] Now, express \( f(t) \) in a form that allows the application of linearity of the Laplace Transform.
4Step 4: Apply Linearity of the Laplace Transform
Use the linearity of the Laplace transform: \( \mathscr{L}\{f(t)\} = \mathscr{L}\left\{\frac{1}{2}\right\} + \mathscr{L}\left\{ \frac{\cos(2t)}{2} \right\} \). Distribute the Laplace transform across the sum of functions.
5Step 5: Apply Laplace Transform to Constant Term
Find the Laplace transform of the constant term: \[ \mathscr{L}\left\{\frac{1}{2}\right\} = \frac{1}{2}\mathscr{L}\{1\} = \frac{1}{2} \cdot \frac{1}{s} = \frac{1}{2s}, \] where \( \mathscr{L}\{1\} = \frac{1}{s} \).
6Step 6: Apply Laplace Transform to Cosine Term
Find the Laplace transform of the cosine term: \[ \mathscr{L}\left\{ \frac{\cos(2t)}{2} \right\} = \frac{1}{2} \cdot \frac{s}{s^2 + 4}, \] where \( \mathscr{L}\{\cos(at)\} = \frac{s}{s^2 + a^2} \) with \( a = 2 \).
7Step 7: Combine Laplace Transforms of Both Terms
Combine the results from Steps 5 and 6: \[ \mathscr{L}\{f(t)\} = \frac{1}{2s} + \frac{s}{2(s^2 + 4)}. \] This expression gives the Laplace Transform of the original function \( \cos^2 t \).
Key Concepts
Trigonometric IdentitiesLinearity of Laplace TransformCosine Function
Trigonometric Identities
Trigonometric identities are powerful tools in mathematics that help simplify complex expressions involving trigonometric functions like sine and cosine. These identities provide relationships between the functions that can transform them into more manageable forms, which is especially useful when solving problems that involve integration or transformation techniques.
In our example, the identity \( \cos^2 t = \frac{1 + \cos(2t)}{2} \) is used. This identity is derived from the double-angle formulas, which are part of a family of identities used to relate angles and their multiples. By applying this identity, the square of the cosine function \( \cos^2 t \) is converted into a combination of a constant term and another cosine function, \( \cos(2t) \).
This transformation is pivotal because it breaks down the squared term into simpler components, allowing for easier manipulation and application of further mathematical processes, such as the Laplace Transform.
In our example, the identity \( \cos^2 t = \frac{1 + \cos(2t)}{2} \) is used. This identity is derived from the double-angle formulas, which are part of a family of identities used to relate angles and their multiples. By applying this identity, the square of the cosine function \( \cos^2 t \) is converted into a combination of a constant term and another cosine function, \( \cos(2t) \).
This transformation is pivotal because it breaks down the squared term into simpler components, allowing for easier manipulation and application of further mathematical processes, such as the Laplace Transform.
Linearity of Laplace Transform
The Laplace Transform is a valuable tool in engineering and physics for solving differential equations. One of its key properties is linearity, which states that the Laplace Transform of a sum is the sum of the Laplace Transforms of its components. Mathematically, this can be expressed as:
In the problem of finding \( \mathscr{L}\{\cos^2 t\} \), after transforming \( \cos^2 t \) using a trigonometric identity into \( \frac{1}{2} + \frac{\cos(2t)}{2} \), we utilize the linearity of the Laplace Transform. We break the function into two parts: the constant \( \frac{1}{2} \) and the scaled cosine function \( \frac{\cos(2t)}{2} \).
This property allows us to transform each component separately and then add the results, significantly simplifying the process of finding the Laplace Transform of more complex functions.
- \( \mathscr{L}\{af(t) + bg(t)\} = a \mathscr{L}\{f(t)\} + b \mathscr{L}\{g(t)\} \)
In the problem of finding \( \mathscr{L}\{\cos^2 t\} \), after transforming \( \cos^2 t \) using a trigonometric identity into \( \frac{1}{2} + \frac{\cos(2t)}{2} \), we utilize the linearity of the Laplace Transform. We break the function into two parts: the constant \( \frac{1}{2} \) and the scaled cosine function \( \frac{\cos(2t)}{2} \).
This property allows us to transform each component separately and then add the results, significantly simplifying the process of finding the Laplace Transform of more complex functions.
Cosine Function
The cosine function \( \cos(t) \) is a basic trigonometric function that represents the horizontal coordinate of a point on the unit circle as it moves around the circle. This function is periodic, repeating its values in a regular cycle, specifically every \( 2\pi \) radians.
Understanding how the cosine function behaves is important for applying transformations like the Laplace Transform. For example, the Laplace Transform of \( \cos(at) \) is particularly straightforward:
In the case of \( \cos(2t) \), the frequency is affected by the term \( 2 \), leading to the specific transformation \( \frac{s}{s^2 + 4} \). Applying this formula directly provides part of the solution when finding the Laplace Transform of expressions involving cosine components. Recognizing these characteristics and transformations helps streamline the process of solving these problems.
Understanding how the cosine function behaves is important for applying transformations like the Laplace Transform. For example, the Laplace Transform of \( \cos(at) \) is particularly straightforward:
- \( \mathscr{L}\{\cos(at)\} = \frac{s}{s^2 + a^2} \)
In the case of \( \cos(2t) \), the frequency is affected by the term \( 2 \), leading to the specific transformation \( \frac{s}{s^2 + 4} \). Applying this formula directly provides part of the solution when finding the Laplace Transform of expressions involving cosine components. Recognizing these characteristics and transformations helps streamline the process of solving these problems.
Other exercises in this chapter
Problem 38
In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{e^{2-t} q(t-2)\right\\} $$
View solution Problem 38
Use the Laplace transform to solve the given initial-value problem. $$ y^{\prime \prime}+9 y=e^{t}, \quad y(0)=0, y^{\prime}(0)=0 $$
View solution Problem 39
Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$ f(t)=t e^{t}+\int_{0}^{t} \tau f(t-\tau) d \tau $$
View solution Problem 39
In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{t^{\boldsymbol{q}} U(t-2)\right\\} $$
View solution