the transform of the hyperbolic functions
The hyperbolic functions cosh and sinh are the exponential cousins of cosine and sine. Where the trig pair describes oscillation, the hyperbolic pair describes the symmetric growth-and-decay you get from combining e^(at) and e^(-at). Their transforms round out the building-block table and look almost identical to the trig ones — with one sign flipped.
By definition cosh(at) = (e^(at) + e^(-at))/2 and sinh(at) = (e^(at) - e^(-at))/2. Since you already know L{e^(at)} = 1/(s - a), linearity does all the work: L{cosh(at)} = (1/2)[1/(s - a) + 1/(s + a)] = s/(s^2 - a^2), and L{sinh(at)} = (1/2)[1/(s - a) - 1/(s + a)] = a/(s^2 - a^2). Compare with the trig results: the only difference is a minus sign in the denominator, s^2 - a^2 instead of s^2 + a^2. That single sign is the whole distinction between oscillation (plus) and exponential blend (minus).
These appear most often the other way around — during inversion. If a partial-fraction step leaves you with s/(s^2 - 4), you should recognize it not as something oscillatory but as cosh(2t). The minus sign in the denominator is the giveaway. Equivalently, s^2 - 4 = (s - 2)(s + 2) factors over the reals, so you could also invert by partial fractions into (1/2)e^(2t) + (1/2)e^(-2t) — which is exactly cosh(2t). Both roads lead to the same place; the hyperbolic shortcut just saves a step.
Invert 3/(s^2 - 16). Here a^2 = 16 so a = 4, and the numerator matches the sinh pattern a/(s^2 - a^2) with a = 4. So 3/(s^2 - 16) = (3/4) [ 4/(s^2 - 16) ] inverts to (3/4) sinh(4t).
A minus in the denominator (s^2 - a^2) means hyperbolic, not oscillatory.
The sign of the denominator is everything: s^2 + a^2 gives oscillation (sin, cos), while s^2 - a^2 gives growth/decay (sinh, cosh). Mixing them up turns a stable wiggle into a runaway exponential.