the unboundedness of complex sine and cosine

On the real line cos x and sin x are the tame, bounded functions: they never leave the band between -1 and 1. It is tempting to carry that intuition into the complex plane. You must not. Complex sine and cosine are wildly unbounded, and seeing why is a good first lesson in how complex extension changes things.

Use cos z = cos x cosh y - i sin x sinh y with z = x + i y. The hyperbolic factors cosh y and sinh y blow up exponentially as |y| grows: cosh y is about (1/2) e^|y| for large |y|. So move straight up the imaginary axis, z = i y, and you get cos(i y) = cosh y, which races off to infinity. A clean estimate is |cos z| >= |sinh y| and |sin z| >= |sinh y|, so both moduli grow at least as fast as (1/2)(e^|y| - 1). The functions are bounded on any horizontal strip of finite height but unbounded on the whole plane.

This is not a pathology; it is forced. Liouville's theorem says a bounded entire function must be constant. Since cos z and sin z are entire and not constant, they cannot be bounded. The same argument explains why no non-constant entire function can stay small. Practically, the unboundedness is exactly the lever that makes residue calculus work: on a large semicircular arc in the upper half-plane you can kill an integrand like e^(i z) f(z) because e^(i z) decays as y grows, but you must be careful, because cos z and sin z themselves grow there.