singleangle

Singleangle is a descriptive term used in some educational and problem-set contexts to refer to methods or problems that rely on a single measured angle to determine other quantities. It is not a formal term in standard mathematical nomenclature, but it appears in textbooks and contest materials to categorize angle-focused problem solving.

In practice, singleangle problems combine one known angle with additional constraints, such as a given side length, parallelism, or a relationship to another figure, to deduce unknown angles or side lengths. The approach often uses trigonometric ratios, similarity, or the law of sines and law of cosines to propagate information from the known angle to the rest of the figure.

In a right triangle, knowing one acute angle and a single side is typically sufficient to determine the remaining sides and the other acute angle via cosine and sine. In non-right triangles, a known angle can still lead to determinations through similarity or trigonometric relationships, but without extra information the solution may be non-unique.

Examples illustrate the idea. A right triangle with angle A = 30° and the leg adjacent to A of length 5 units yields the hypotenuse as 5 / cos 30°, and the opposite leg as 5 tan 30°. An isosceles triangle with apex angle 40° and a known base length b allows computing the equal sides using the law of cosines.

Applications include teaching problem design, surveying, and computer vision, where a single angle can anchor measurements. Limitations include potential measurement errors and ambiguity when only one angle is insufficient to guarantee a unique solution.

See also: Trigonometry, Law of Sines, Law of Cosines, Angle, Similarity.