Trigonometry and phase: turning rotation into a wave

An angle in radians is arc length divided by circle radius. The ratio is dimensionless, and a complete turn corresponds to $2\pi$ radians. Measuring angles this way is natural for calculus because the arc length on a unit circle equals the angle numerically. Degrees remain useful for communication, but formulas involving derivatives of sine and cosine assume radians unless an explicit conversion factor is included.

On a circle of radius one, cosine is the horizontal coordinate and sine the vertical coordinate. Negative values represent points on the opposite side of an axis; they are not negative distances from the origin. The circle equation constrains the coordinates. Knowing one coordinate does not always determine the angle, because points in different quadrants can share a sine or cosine value.

Amplitude sets the size of an excursion away from the central value. A period is the time needed for a complete repeat, and frequency counts repeats per unit time. Angular frequency counts radians of rotation per unit time. Keeping these concepts separate prevents a common factor-of-$2\pi$ error. High amplitude does not imply high frequency, and neither alone specifies a wave's initial position in its cycle.

Phase specifies that position within a cycle. A phase offset shifts the pattern relative to the chosen time origin. Only relative phase matters when comparing two oscillations measured with the same clock. Adding $2\pi$ to a phase gives the same sine or cosine, so different numerical angles can describe the same cycle position. For positive angular frequency, a positive offset in cosine moves a given peak earlier in time.

Two equal signals in phase add to twice the amplitude at every instant. If their phases differ by $\pi$, one is the negative of the other and their sum is zero. Cancellation describes the sum at the same location and time under a linear addition rule. It does not mean either source disappears or that energy accounting can be ignored in a physical interference experiment.

A sinusoid is an ideal building block. Real signals can contain many frequencies, changing amplitudes, noise, or abrupt transitions. A single frequency model is still useful when an experiment is close to periodic or when a complicated signal is decomposed into components. Before interpreting a wave picture, ask which axis is time and whether the animation represents a point's oscillation or a pattern traveling through space.