Building trigonometric systems for circle quadrilaterals in a circle and its applications

Abstract

This study is to construct trigonometric systems for a cyclic quadrilateral in a given circle. Brahmagupta's formula for calculating the area value of a cyclic quadrilateral is introduced and proven, in detail. Then, trigonometric formulas are built to compute the angle value of a circle quadrilateral in a circle such as the sine, the cosine, and other related height formulas. Besides, some formulas for calculating the area value of triangles, parallelograms, rhombus, rectangles and squares are described as a direct application to the formula of the Brahmagupta-type. Some illustrative examples for problems related to a circle quadrilateral in a circle are proposed to apply the obtained results.