Hang a weight on a string, set it swinging. Tape a marker to the weight. Slowly pull a strip of paper underneath. The marker traces — a sine wave.
Why a sine?
A pendulum's horizontal position oscillates between left and right with smooth turning at each extreme. That's exactly what sin(t) does: oscillate between −1 and +1 forever.
Almost everything that vibrates — strings, springs, AC current, sound — moves as a sine wave (or a sum of them). Fourier proved it.