Question

Consider \( f(t) = \cos(at) \), where \( a \) is a real constant. The Laplace transform of \( f(t) \) is _________.

• A. \( \frac{a}{s^2 + a^2} \)
• B. \( \frac{s}{s^2 + a^2} \)
• C. \( \frac{a}{s^2 - a^2} \)
• D. \( \frac{s}{s^2 - a^2} \)

Hint:

The Laplace transform of \( \cos(at) \) is given by \( \frac{s}{s^2 + a^2} \). Always refer to standard transform tables for common functions like trigonometric functions.

Answer 1

The Correct Option is B

To find the Laplace transform of \( f(t) = \cos(at) \), we use the standard Laplace transform formula for \( \cos(at) \): \[ \mathcal{L}\left\{ \cos(at) \right\} = \frac{s}{s^2 + a^2} \] Step 1: Laplace transform of cosine function: The formula for the Laplace transform of \( \cos(at) \) is: \[ \mathcal{L}\left\{ \cos(at) \right\} = \frac{s}{s^2 + a^2} \] Step 2: Explanation of options:
Option (A): \( \frac{a}{s^2 + a^2} \) — Incorrect. This is not the correct formula for the Laplace transform of \( \cos(at) \).
Option (B): \( \frac{s}{s^2 + a^2} \) — Correct. This is the standard Laplace transform of \( \cos(at) \).
Option (C): \( \frac{a}{s^2 - a^2} \) — Incorrect. This is not the correct formula for the Laplace transform of \( \cos(at) \).
Option (D): \( \frac{s}{s^2 - a^2} \) — Incorrect. This formula is not for \( \cos(at) \); it is for a different type of function.
Conclusion: The correct Laplace transform of \( \cos(at) \) is \( \frac{s}{s^2 + a^2} \), which corresponds to option (B).