How is the Laplace transform of the function f(t) = sin(at) derived in the chapter?

The Laplace transform of the function f(t) = sin(at) is derived using the definition of the Laplace transform, which involves integrating sin(at) multiplied by e^(-st) from 0 to infinity. This can be simplified by recognizing that sin(at) is the imaginary part of e^(iat), allowing for a more straightforward calculation.

To derive the Laplace transform of f(t) = sin(at), we start with the definition of the Laplace transform, L{f(t)} = ∫[0 to ∞] f(t)e^(-st) dt. Substituting f(t) = sin(at), we have L{sin(at)} = ∫[0 to ∞] sin(at)e^(-st) dt. Instead of performing integration by parts, we can express sin(at) as the imaginary part of e^(iat), which simplifies the process. Thus, we can write L{sin(at)} = Im{L{e^(iat)}} = Im{∫[0 to ∞] e^(iat)e^(-st) dt} = ∫[0 to ∞] e^(-(s - ia)t) dt, leading to the result after rationalizing the denominator.