Trigonometry is not the work of one man or nation. Ancient Egyptians and Babylonians had developed theorems on the ratios of the sides of similar triangles before trigonometry was formalized as a subdivision of mathematics. Egyptians used trigonometry to their benefit of landscaping and building pyramids. The Babylonian astronomers related trigonometric function to the arcs of circles and the lengths of chords subtending their arcs. Trigonometry is taken from two Greek words trigonon meaning triangle and metria meaning measure.
The goal or objective of this project was to find out why the trigonometric functions were defined as the ratio
of two sides of the triangle with sine being opposite over the hypothenuse, cosine being adjacent over the hypothenuse,
and the tangent being opposite over adjacent. The first fact that I discovered was that the trigonometric functions
were defined by the lenghts of the sides of the triangles made by the chord and secant line of the
Unit Circle.
Hipparchus, Astronomer, and father of Trigonometry, made a table of chords closely related to the trigonometric table for
the sine function. From this fact we can conclude that the chord of a circle and the sine function are related. How?
The Egyptians put some light on this relationship by using chords from the unit circle to measure the sides of the pyramids
they were building. Since the pyramids were triangular, they inscribed a triangel inside a circle. One corner was at the origin
of the circle and the other two corners touched the circumference of the circle. The segment the two corners made
became a chord of the circle. Then they drived the formula to measure the chord. The two videos below explain their
process.
It can be mathematically proven that there is a relationship between the sine function and the chord of a circle. When we prove this relationship we also prove that the sine function can be defined as the ratio of the opposite side over the hypotenuse. If we can prove the ratio for the sine function then we can conclude there are proofs for the other 5 trig functions being defined as ratios of the two sides of a triangle.
Ransom, M, and K Mathis. “Origins of the Names of Trigonometric Functions.” Origins of the Names of Trigonometric Functions
, 2003, www.algebralab.org/lessons/lesson.aspx?file=Trigonometry_TrigNameOrigins.xml.
Adamek, T., Penkalski, K., & Valentine, G. (2005). The History Of Trigonometry. History of Mathematics,
01(640), 1–19. Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.120.7391&rep=rep1&type=pdf
An Introduction To Trigonometry. (2016). Retrieved from https://www.youtube.com/watch?v=Ls8mwrbeg7k
History of Trigonometry. (2013). Retrieved from https://www.youtube.com/watch?v=q5zWIp23jB4
Origin of Trigonometry. (2015). Retrieved from https://www.youtube.com/watch?v=WPz1OwdkfgY
Origins of Trigonometry. (n.d.). Retrieved from https://www.youtube.com/watch?v=fC_NGpz7xQM
Rick, D. (1999, October 27). Origin of the Terms Sine, Cosine, Tangent, etc.
Retrieved October 10, 2019, from http://mathforum.org/library/drmath/view/52578.html.
buscherini, S. (2010). The Table of Chords and Greek Trigonometry. Conservation Science in Cultural Heritage.
Retrieved from https://conservation-science.unibo.it/article/view/2313/1697
The Very Early History of Trigonometry . (n.d.). Retrieved from https://people.sc.fsu.edu/~dduke/earlytrig12.pdf
Developing the Radian Concept Understanding and the Historical Point of View . (2008).
Retrieved from http://math.unipa.it/~grim/Quad18_Kupkova_08.pdf
Calculating Chord Length. (2017). Retrieved from https://www.youtube.com/watch?v=mgKbpTtDxSk