Rainbow Trigonometry

A phase shift in a sinusoidal signal moves its peaks and troughs along the input axis without changing amplitude or period. The general form \sin(x + \varphi) makes \varphi the horizontal offset, and phase is a central concept in wave physics, signal processing, and audio engineering. Two signals can interfere constructively, destructively, or somewhere in between depending entirely on their relative phase, which is why phase relationships govern everything from antenna arrays to noise-cancelling headphones.

The plot overlays six sine waves of the form \sin(x + k/6) for k = 0 through 5, stroked in red, orange, yellow, green, blue, and purple. Because amplitude and frequency are held constant, the difference between any two curves is a pure horizontal shift, and the points where neighboring curves cross trace the identity \sin(a) - \sin(b) = 2\cos((a+b)/2)\sin((a-b)/2). Reading from one color to the next gives a direct visual feel for what "phase offset" really means.