Compute the Laplace transform of sin t (assume unilateral transform and zero initial conditions).

Introduction / Context: The Laplace transform converts time-domain signals to the s-domain, simplifying differential equations to algebraic forms. Trigonometric inputs such as sin t and cos t are classics, and remembering their transforms is essential for solving control and circuit problems quickly.

Given Data / Assumptions:

• Function: f(t) = sin t for t ≥ 0.
• Unilateral Laplace transform definition is used.
• Zero initial conditions; standard transform tables apply.

Concept / Approach: The standard Laplace transform of sin(ω t) is ω / (s^2 + ω^2). Setting ω = 1 yields 1 / (s^2 + 1). This can also be derived via integration using Euler’s identities or by differentiating the transform of cos t with respect to s.

Step-by-Step Solution:

Verification / Alternative check: Differentiate the transform pair L{cos t} = s / (s^2 + 1) with respect to s and apply relationships between sin and cos to reconfirm the result.

Why Other Options Are Wrong:

Common Pitfalls: Swapping the transforms of sin t and cos t; remember sin has numerator ω, cos has numerator s.

Final Answer: 1 / (s^2 + 1)