# Trigonometry Research Paper

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Abstract— The history of trigonometry extends from the beginning of the development of mathematics in Greece to the enlightenment in Europe. While it has evolved from the original chord function to a plethora of functions and identities throughout the ages, many of the identities were already known in antiquity and utilized to measure distance and location in the skies. This paper provides a short review of the development of the sine and a short review of the development of the remaining trigonometric functions. The paper ends with an instructional suggestion to teach the fundamental and Pythagorean trigonometric identities using a diagram composed of the unit circle and similar triangles that will allow high school students to discover the …show more content…
Mathematicians of antiquity were concerned with the measurement of angles in the skies, and the geometry invented to deal with this problem was called spherics and was studied as part of the quadrivium of study [1]. While spherics became elliptic geometry, trigonometry evolved in its own path. Any discussion of trigonometry, however, must discuss the concept of angle first. Degree measurement for angles was already in use by the Babylonians in the year 300 B.C.E. This civilization was the first to assign coordinates to stars.
The birth of trigonometry would occur around 140 B.C. when the Greek mathematician Hipparchus produced the first table of chords. For the Greeks, it was important to know the ratio of any chord on the circle to the diameter of the circle for astronomical purposes. While the Greeks did not relate the table of chords to the modern sine function, the ratio of the length of the chord of a circle to the diameter is always equal to the sine of half of the angle subtending the given chord. Therefore, the tables of chords are also known today as the first tables of the sine
in a similar manner to Hipparchus, Ptolemy innovated upon Hipparchus’ method by inscribing regular polygons inside a circle and calculating the resulting chords from each polygon. By combining this with a method to find the chord subtended by half the arc of a known chord, Ptolemy calculated chord lengths accurate to six decimal places [2]. Ptolemy and his predecessors already knew about the Pythagorean identity, sin^2⁡x+cos^2⁡x=1 expressed as chords of a circle instead of our modern notation. Ptolemy also knew the formulas for the sine of the sum of two angles and the law of sines expressed with chords

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