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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]
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
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 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 of the ...
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 ...
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.
Fig. 1a – Sine and cosine of an angle θ defined using the unit circle Indication of the sign and amount of key angles according to rotation direction Trigonometric ratios can also be represented using the unit circle , which is the circle of radius 1 centered at the origin in the plane. [ 37 ]
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 ...
Bhāskara (c. 600 – c. 680) (commonly called Bhāskara I to avoid confusion with the 12th-century mathematician Bhāskara II) was a 7th-century Indian mathematician and astronomer who was the first to write numbers in the Hindu–Arabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's ...