Topics in Trigonometry!

Trigonometry- definition  The word trigonometry comes from Ancient Greek: τρίγωνο" (triangle) + "μετρε ῖ ν" (to measure). “Triangle measuring”  This was it’s given name because trigonometry had mainly been used for “triangle measuring” at its time.

History  There is not much real history about trigonometry; it’s something that was discovered and developed over many years.  Uses of trigonometry date back to times of Ancient Egypt and Greece -The Egyptians used a basic form of trigonometry to build their pyramids. They wondered that if If a pyramid is 250 cubits high and the side of its base 360 cubits long, what would its “seked” be? Seked means the length of the slope, which would be the hypotenuse. They used a primitive form of trigonometry for this.

Trig and the Pyramids

The Father of Trigonometry, Hipparchus  Hipparchus- 190 BC- 120 BC  Hipparchus was a Greek astronomer who was the first to make accurate models for the motion of the Sun and Moon that still survive today.  His greatest contribution to trigonometry was constructing the first trigonometric table  He spent most of his life working on astronomy. Trig was an outocme of his astronomy.

Development- the Greeks  After Hipparchus passed, the Greeks were still the major power in mathematics.  Menelaus made theories on spherical triangles and chords.  Ptolemy continued on Hipparchus’ work on chords and made some important formulas and theorems, the most important being the half-angle formula.

Menelaus of AlexandriaPtolemy Hipparchus, the “Father of Trigonometry”

Development- the Islamic World  The Persians and Arabs converted the early trigonometry works for the medieval Islamic world.  The most important contribution to basic trigonometry by the Muslims:  Muhammad ibn Mūsā al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents. He was also a pioneer in spherical trigonometry, picking up after Menelaus of Greece. Besides his enormous contributions to trig, he is widely known as the creator of algebra.  In 830, Habash al-Hasib al-Marwazi produced the first table of cotangents. Muhammad ibn Jābir al-Harrānī al-Battānī discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. Habash al-Hasib al-Marwazi Muhammad ibn Jābir al-Harrānī al-Battānī

Muhammad ibn Mūsā al-Khwārizmī Habash al-Hasib al-Marwazi Muhammad ibn Jābir al-Harrānī al-Battānī

Development- Europe  The biggest contribution to our basic trigonometry came from Georg Joachim Rheticus.  He was the first in Europe to apply trigonometric functions directly in terms of right triangles instead of circles, and created tables for all six trigonometric functions.

Applications of Trig in the Modern World  The world has used trigonometry and trigonometric functions in unlimited number of ways. For example, triangulation is used in astronomy to find the distance between nearby stars. Also, triangulation is used to measure distances between major landmarks. Trigonometry is also used in satellite navigation systems. Cosine and sine functions are key to the theory of periodic functions like those that explains sound and light waves. Jobs that incorporate trigonometric functions or trigonometry include almost all sciences, computer graphics and gaming, pharmacy, number theory, and electronics.

Diagrams A basic right triangle with angle theta (Θ) SOH-CAH-TOA is a common tool to remember the 3 basic trig functions.

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