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Received Nov 15, 2017; Revised Feb 26, 2018; Accepted Apr 11, 2018
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1. Introduction
The term “Trigonometry” was first coined as the title of a book (Trigonometria), which translates to “triangle’s measurement.” Although trigonometry is now taught with an emphasis on right triangles, its origin goes back to an era where it was used to determine the positions of celestial bodies and the distances between them and to understand the concept of chords in circles. Euclidean geometry, that is, planar geometry, deals with two-dimensional figures. The study of planar geometry provides constructions for planar figures, their properties, and the relationships between points, lines, and figures. The planar figure formed when a point traces a path at a fixed distance with respect to another fixed point called a circle, where the circle divides a plane into interior and exterior regions. Various theorems and properties on circles have been developed since time immemorial.
Gradually, researchers have developed theories on intersecting circles, which led to divergent properties between circular triangles and spherical triangles. The introduction of fuzzy sets and systems by Zadeh [1] changed the face of research in trigonometry. Fuzzy trigonometry was introduced by Buckley and Eslami [2], wherein continuous fuzzy numbers and sets were defined using the principle of extension. This method formed the basis of fuzzy trigonometry but failed to satisfy many criteria and identities. Furthermore, Ress [3] developed an approach for mapping standard trigonometric functions into the fuzzy realm. Using these modified fuzzy trigonometric functions, the proofs of a few inverse trigonometric identities, in addition to the standard identities, were given. A breakthrough in the study of fuzzy trigonometry was achieved by Liu et al. [4], with an aim of connecting symbolic cognitive functions to qualitative functions. The basic identities were satisfied, but a few properties could not be achieved. Ghosh and Chakraborty [5] proposed two methodologies for describing a fuzzy circle. The first methodology defines a circle as a set of points which are equidistant from a fixed point. The second methodology describes a circle using three fuzzy points. The definitions using...