Any reasonable periodic function can be written as a sum of sines and cosines — its Fourier series.
far apart → no slow beat; the sum is a busy mixture of both — that's how a chord or a vowel sound is built
Adding waves to build a shape
The curve above is sin x + (1/3)sin 3x + (1/5)sin 5x + … — just a few odd-harmonic sine waves stacked up, and already it's flattening into a square wave. Add infinitely many terms and the wiggles vanish (almost — see the overshoot near the jumps, the famous Gibbs phenomenon). Every periodic signal is a recipe of sines and cosines like this.
The coefficients aₙ, bₙ are found by integrating f against cos nx and sin nx over one period — each one extracts 'how much' of that frequency is present.
MP3 and JPEG compression, noise-cancelling headphones, MRI reconstruction, and Wi-Fi signal processing all run on Fourier analysis — breaking a signal into frequencies, tweaking them, rebuilding.
The square-wave series uses only sin x, sin 3x, sin 5x, … Why no even harmonics or cosines?
Fourier series power signal processing, audio compression (MP3), JPEG image compression, and much of modern engineering.