phaseangle

Phase angle, often called the phase or argument, is a measurement of orientation or timing used across mathematics, physics, and engineering. It describes the rotation of a complex quantity or a sinusoidal signal relative to a reference axis or instant.

In mathematics, a complex number z = a + bi can be written as z = r e^{iθ}, where r = |z| is the modulus and θ = Arg(z) is the phase angle. The phase angle is the angle from the positive real axis to the vector representing z in the complex plane. It is defined modulo 2π and is typically taken as the principal value between −π and π.

In signal processing, the phase angle is the offset φ in a cosine wave x(t) = A cos(ωt + φ). For two signals, the phase difference Δφ is the difference between their phase angles and governs their relative timing and constructively or destructively interfering behavior.

In astronomy, the phase angle α is the angle Sun–Object–Observer. It describes the portion of the object's illuminated side visible to the observer and affects the observed brightness and the shape of the phase curve for planets, moons, asteroids, and comets.

In optics and scattering, phase angle can also denote the angle between the incident and scattered directions. It influences the observed intensity through phase functions that describe how light is scattered by a medium or surface.

Overall, the phase angle is a fundamental parameter that links geometry, timing, and illumination across multiple disciplines.