Question

Assuming $s>0$, the Laplace transform for $f(x) = \sin(ax)$ is

• A. $\dfrac{a}{s^2 + a^2}$
• B. $\dfrac{s}{s^2 + a^2}$
• C. $\dfrac{a}{s^2 - a^2}$
• D. $\dfrac{s}{s^2 - a^2}$

Hint:

Remember: $\sin(ax)$ gives $a$ in the numerator, $\cos(ax)$ gives $s$. Both have $s^2 + a^2$ in the denominator.

Answer 1

The Correct Option is A

The Laplace transform of a sine function is well known: \[ \mathcal{L}\{\sin(ax)\} = \int_0^\infty e^{-sx} \sin(ax)\, dx = \frac{a}{s^2 + a^2}, \qquad s>0. \] Options (C) and (D) correspond to hyperbolic sine ($\sinh$), and option (B) is the Laplace transform of $\cos(ax)$. Thus, the correct transform is: \[ \mathcal{L}\{\sin(ax)\} = \frac{a}{s^2 + a^2}. \] Final Answer: $\dfrac{a}{s^2 + a^2}$