Mathematicians Who Contributed in Trigonometry

MATHEMATICIANS WHO CONTRIBUTED IN TRIGONOMETRY ü Hipparchus – created the science of trigonometry ü Ptolemy – calculated the table of chords called the almagest ü Hindus – made a table of half chords or sine ü Al Battani – calculated a table of cotangents ü Arab – invented the tangent ü Johannes Miller – wrote the first complete European treaties on trigonometry ü Menelaus – gave the theorem that the sum of the spherical triangle is = 180 ü Briggs – made logarithmic tables of all kinds ü Leonhard Euler – treated Trigonometry as a branch of analysis The motivation for developing trigonometry was to solve spherical triangles. This was needed to make astronomical calculations for use in astrology. They first developed plane trigonometry to solve plane triangles. This was generalized to spherical trigonometry which they applied to solve spherical triangles. •

Hipparchus (190 – 120 BCE)

Hipparchus lived in Nicaea. He defined the chord function, derived some of its properties and constructed a table of chords for angles that are multiples of 7.5° using a circle of radius R = 60 × 360/(2π).This his motivation for choosing this value of R. In this circle, the circumference is 360 times 60. Therefore, the length of the arc of a sector equals the angle of the sector measured in minutes. Hipparchus used crd α and crd (180 – α) as his trigonometric functions. Hipparchus used the following properties in his computations: [crd (180 – α)]2 = 4R2 – (crd α)2 which is equivalent to the “ Pythagorean identity”, cos2 α/2 = 1 – sin2 α/2. •

Menelaus (ca 100 CE)

Menelaus of  Alexandria wrote Chords in a Circle consisting of six books which contain many advances in spherical geometry and some of the basic formulas used to solve spherical triangles. Here are three of his results. 1. The sum of two sides of a spherical triangle is greater than the third side while the sum of its angles is greater than 180°. 2. If two spherical triangles have the same angles, then they are congruent. 3. Menelaus’ Theorem was used by the Greeks to prove the identities.

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Ptolemy (100 – 178 CE)

Ptolemy lived in Alexandria. His work on astronomy included a sophisticated geometric model of the solar system as well as methods of computation of celestial phenomena. To do this, he extended the work of Hipparchus on trigonometry and the work of Menelaus on spherical trigonometry which he applied to solve the spherical triangles that arise in astronomy. His book, the Almagest, was the basis for astronomy in the Hellenist world, in the Islam world and in Europe until the Renaissance. •

Liu Hui (3rd century CE)

He wrote a collection of nine problems called Haidao Suanjing (Sea Island Mathematical Manual) to compute lengths indirectly. Liu Hui’s solutions use areas. However, later solutions used similar triangles and led to the introduction of the tangent function to make calculations. •

Rheticus (1542)

He is a student of Copernicus, defined the sine as a function of an angle α rather than a function of the corresponding arc of a circle. He used the Indian sine function ½ crd 2α = Rsin α. •

Aryabhata (476–550 CE)

He was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are in the Aryabhatiya and the Arya-siddhanta. Aryabhata's treatise, contains table of sines calculating the approximate values at intervals of 90°/24 = 3° 45'. He also introduced the versine (versin = 1 - cosine) into trigonometry. Aryabhatta used the words jya for sine, kojya for cosine, utkrama-jya for versine, and otkram jya for inverse sine. The words jya and kojya eventually became sine and cosine respectively. •

Abu’l Wafa of Baghdad

The idea of setting R = 1 in the chord function, which leads to the modern trigonometric functions, originated in the 10th century with Abu’l Wafa of Baghdad and was also suggested by Rawlinson of Oxford ca 1660. However, this idea was ignored until 1748 when it was reintroduced simultaneously in the texts of Leonhard Euler and Thomas Simpson.