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

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

2008-04-22

arxiv.org/abs/0804.3600v1

Mathematics

General Mathematics

16 pages

Scientific paper

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.

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