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For the second one, the text states: "We multiply the sine of each of the two arcs by the cosine of the other minutes. If we want the sine of the sum, we add the products, if we want the sine of the difference, we take their difference". [45] He also discovered the law of sines for spherical trigonometry: [41]
In mathematics, sine and cosine are trigonometric functions of an angle.The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that ...
He was the first to use the abbreviations 'sin', 'cos' and 'tan' for the trigonometric functions in a treatise. [1] Girard was the first to state, in 1625, that each prime of the form 1 mod 4 is the sum of two squares. [3] (See Fermat's theorem on sums of two squares.) It was said that he was quiet-natured and, unlike most mathematicians, did ...
The sine and cosine functions are fundamental to the theory of periodic functions, [63] such as those that describe sound and light waves. Fourier discovered that every continuous , periodic function could be described as an infinite sum of trigonometric functions.
Among his many contributions, he discovered infinite series for the trigonometric functions of sine, cosine, arctangent, and many methods for calculating the circumference of a circle. One of Madhava's series is known from the text Yuktibhāṣā , which contains the derivation and proof of the power series for inverse tangent , discovered by ...
In mathematics, a Madhava series is one of the three Taylor series expansions for the sine, cosine, and arctangent functions discovered in 14th or 15th century in Kerala, India by the mathematician and astronomer Madhava of Sangamagrama (c. 1350 – c. 1425) or his followers in the Kerala school of astronomy and mathematics. [1]
Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics. [8] The now lost tables of Hipparchus (c. 190 BC – c. 120 BC) and Menelaus (c. 70–140 CE) and those of Ptolemy (c. AD 90 – c. 168) were all tables of chords and not of half-chords. [8] Āryabhaṭa's table remained as the standard sine table of ...
4th to 5th centuries: The modern fundamental trigonometric functions, sine and cosine, are described in the Siddhantas of India. [75] This formulation of trigonometry is an improvement over the earlier Greek functions, in that it lends itself more seamlessly to polar co-ordinates and the later complex interpretation of the trigonometric functions.