The equation asinx+bcosx=c and a family of cyclic Heron quadrilaterals

TL;DR

This paper solves the trigonometric equation asinx+bcosx=c, explores special cases involving right triangles, and constructs an infinite family of cyclic Heron quadrilaterals with integer sides, diagonals, and areas.

Contribution

It provides a general solution to the equation, characterizes cyclic quadrilaterals with specific properties, and introduces a new family of integer-sided Heron quadrilaterals.

Findings

Explicit general solution to asinx+bcosx=c.

Construction of cyclic Heron quadrilaterals with integer parameters.

Identification of a three-parameter infinite family of such quadrilaterals.

Abstract

In the beginning of this paper, we present the general solution to the trigonometric equation asinx+bcosx=c. After that, we focus on the case when a^2+b^2=c^2. In this case, the general solution is expressed in terms of the acute angle theta which satisfies tan(theta)=a/b+c .From the right trianle with leglengths a and b, and hypotenuse length c, we construct a cyclic quadrilateral within which the angle theta is illustrated.Then we examine the case when a,b,and c are integers;and we derive or construct a family of cyclic Heron quadrilaterals. These are quadrilaterals with integer sidelengths or edges, integer diagonal lengths, and integer area.Also these quadrilaterals have their four vertices lying on a circle. The said family of quadrilaterals, is a three parameter infinite family.