scaling property of the Laplace transform
Suppose you run a process faster or slower — you speed up the clock, compressing the same behavior into less time. The scaling property tells you exactly how that time-rescaling shows up in the transform: stretching or shrinking the time axis stretches or shrinks the s-axis the opposite way, and rescales the height to keep the books balanced.
The rule is: if L{f(t)} = F(s), then for a positive constant a, L{f(a t)} = (1/a) F(s/a). Replacing t by a t (running the process a times faster) replaces s by s/a in the transform and multiplies by 1/a. The 1/a factor comes from the change of variable inside the defining integral: substituting u = a t shrinks the differential by 1/a. The reciprocal relationship between time and the s-variable is the same inverse-scaling you meet in every transform: faster in time means spread out in frequency.
This property is less heavily used than the shifting theorems but it is conceptually important and a genuine labor-saver: once you know one transform in a family, you get the whole scaled family for free, without re-integrating. It also underlies dimensional reasoning — checking that a transform has consistent units under a change of time scale — and it is the Laplace cousin of the more famous Fourier scaling, where the same compress-in-time, stretch-in-frequency trade-off governs the bandwidth of a signal.
Knowing L{sin(t)} = 1/(s^2 + 1), the scaling rule gives L{sin(omega t)} = (1/omega) [ 1/((s/omega)^2 + 1) ] = omega/(s^2 + omega^2), matching the table.
The scaling rule generates the general sine transform from the unit-frequency case with no new integration.
The constant a must be positive; for the one-sided transform a negative scaling would reverse time and pull in the part of the function before t = 0, which the one-sided integral does not see, so the simple rule no longer applies.