the transform of sine and cosine

Oscillation is everywhere — a vibrating string, an alternating current, a swinging pendulum — and the language of oscillation is sine and cosine. So the transforms of sin and cos are essential table entries, and they carry the telltale s^2 + a^2 that signals 'something is wiggling at frequency a.'

The two rules are L{sin(at)} = a / (s^2 + a^2) and L{cos(at)} = s / (s^2 + a^2). They share the same denominator s^2 + a^2; only the numerator distinguishes them — a bare a for sine, an s for cosine. The cleanest way to see them is through Euler's formula: e^(iat) = cos(at) + i sin(at), so L{e^(iat)} = 1/(s - ia). Splitting that single complex transform into its real and imaginary parts hands you both trig transforms at once, with the denominator (s - ia)(s + ia) = s^2 + a^2 falling out automatically. The factor a inside the angle is the frequency, and it is exactly the a^2 you see in the denominator.

When you invert and meet s^2 + a^2 in a denominator, that is your cue: the solution oscillates. A term s/(s^2 + 9) becomes cos(3t); a term 3/(s^2 + 9) becomes sin(3t). This is the s-domain signature of vibration, and it is why solving a spring-mass equation by Laplace naturally produces sines and cosines at the natural frequency — the same answer the characteristic-equation method gives, reached by algebra instead of guesswork.