Let x(t) = exp(st) where s = σ + jΩ.
|x(t)| = exp(σ t)
∠x(t) = Ω t
x(t) forms a spiral in the s-plane.
σ < 0 : spiral in
σ > 0 : spiral out
σ = 0 : no spiral (circle)
Ω < 0 : spiral clockwise
Ω > 0 : spiral counter-clockwise
Ω = 0 : no spiral (trace real axis)
With that, it's obvious why
cos(Ωt) = (1/2) exp(jΩt) + (1/2) exp(-jΩt)
It's just the sum of two opposite circular
"spirals" whose imaginary parts cancel each
other out, tracing out the real axis from
+1 to -1.
That's also why the Laplace or Fourier
transform of cos(Ωt) is two "spikes" in
the frequency domain -- because it's just
the sum of two complex exponentials.