Semiangle

Semiangle is a term used to describe half of a given angle. In geometry and trigonometry, if an angle is θ, its semiangle is θ/2. The concept is often encountered when describing cones, beams, or angular spread, where the semiangle represents the angle between the axis of symmetry and a boundary ray. In optics and engineering, the semiangle of a cone or beam is the angle from the axis to the surface, i.e., half the apex angle of the cone.

In trigonometry, semi-angle formulas arise from the half-angle identities. For any angle θ,

- sin(θ/2) = ±√[(1 − cos θ)/2],

- cos(θ/2) = ±√[(1 + cos θ)/2],

- tan(θ/2) = sin θ / (1 + cos θ) or (1 − cos θ) / sin θ,

with the signs chosen according to the quadrant of θ/2. These formulas are useful in simplifying expressions and in integration or solving trigonometric equations involving half-angles.

Applications and usage vary by field. In geometry, the apex of a cone or the opening of a circular sector can be described by its semiangle. In optics and photonics, the semiangle characterizes the divergence of a beam or the emission cone of a lamp or laser. The term is not always used consistently; some contexts prefer “half-angle” or “aperture angle,” but semiangle remains a common shorthand for half of an angle.