Laplace transform of cos(t)
Understand the Problem
The question is asking for the Laplace transform of the cosine function, which involves applying the definition of the Laplace transform to the function cos(t). The approach includes using the formula for the Laplace transform and possibly identifying any parameters.
Answer
$$ \mathcal{L}\{\cos(t)\} = \frac{s}{s^2 + 1} $$
Answer for screen readers
The Laplace transform of $\cos(t)$ is given by: $$ \mathcal{L}{\cos(t)} = \frac{s}{s^2 + 1} $$
Steps to Solve
- Identify the Definition of Laplace Transform
The Laplace transform of a function $f(t)$ is defined as: $$ \mathcal{L}{f(t)} = \int_0^\infty e^{-st} f(t) , dt $$
- Substitute Cosine Function
For the function $f(t) = \cos(t)$, we substitute this into the Laplace transform formula: $$ \mathcal{L}{\cos(t)} = \int_0^\infty e^{-st} \cos(t) , dt $$
- Use Integration by Parts
To solve the integral, we can use integration by parts or the standard result known for the Laplace transform of cosine. The result is: $$ \mathcal{L}{\cos(at)} = \frac{s}{s^2 + a^2} $$ For our case, $a = 1$.
- Apply the Formula
Now we substitute $a = 1$ into the formula: $$ \mathcal{L}{\cos(t)} = \frac{s}{s^2 + 1} $$
- Conclusion
Thus, the Laplace transform of $\cos(t)$ can be expressed as: $$ \mathcal{L}{\cos(t)} = \frac{s}{s^2 + 1} $$
The Laplace transform of $\cos(t)$ is given by: $$ \mathcal{L}{\cos(t)} = \frac{s}{s^2 + 1} $$
More Information
The Laplace transform is widely used in engineering and physics for analyzing linear dynamic systems. It transforms a function of time into a function of a complex variable, allowing easier manipulation and solving of differential equations.
Tips
- Confusing the parameters for the cosine function; remember $a$ in $\cos(at)$ represents the frequency.
- Forgetting to adjust the formula for different values of $a$; always use the correct transformation based on the function.
