Question:
Consider \( f(t) = \cos(at) \), where \( a \) is a real constant. The Laplace transform of \( f(t) \) is _________.
Consider \( f(t) = \cos(at) \), where \( a \) is a real constant. The Laplace transform of \( f(t) \) is _________.
Show 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.
Updated On: Apr 25, 2025
- \( \frac{a}{s^2 + a^2} \)
- \( \frac{s}{s^2 + a^2} \)
- \( \frac{a}{s^2 - a^2} \)
- \( \frac{s}{s^2 - a^2} \)
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The Correct Option is B
Solution and Explanation
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).
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).
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