Nonspecialangle

Nonspecialangle is an informal label used in geometry and trigonometry to refer to an angle that does not have one of the commonly recognized “special” properties or appearances in problems. The term is not a standard mathematical term, but it is sometimes used in instructional materials, contest problems, or online discussions to emphasize a generic or arbitrary angle without relying on known special cases.

In this context, a “special angle” or special-angle value typically means an angle with a well-known exact trigonometric value or a frequently occurring configuration, such as zero or straight angles and common acute angles like 30°, 45°, 60°, and 90° (together with their radian equivalents). Angles that do not belong to such a set, and whose sine, cosine, or tangent values are not expressible by simple surds, are often considered nonspecial. For example, an angle around 37° is frequently treated as nonspecial because its exact trigonometric values do not simplify to common closed forms.

Usage and caveats: nonspecialangle is mainly a descriptive aid in problem statements to indicate a general case. Results stated for nonspecialangles are not guaranteed to extend to every possible angle, since some properties may fail for certain special angles. When a theorem or identity is tested, authors may separately verify that the result holds for all special angles, with the nonspecialangle case following from a general argument.

See also: generic angle, arbitrary angle, nondegenerate angle.