In the second century B.C., Hipparchus derived a trigonometric table measuring chord lengths of a circle having a fixed radius. Hipparchus built the values in increasing degrees, beginning with 71 and ending with 180, incrementing in units of 71 degrees. In the second century A.D., Ptolemy defined Hipparchus' value for the radius as 60 and created a table of chords incrementing one degree, from 0 degrees to 180 degrees. This table of chords also showed how to find unknown parts of triangles from given parts.
In the sixth century, India based its trigonometry on the sine function, which was the length of the side opposite the angle in a right triangle of a specific hypotenuse instead of a ratio. The Indians used various values for the hypotenuse. They built sine tables from these functions and later introduced a cosine function and tables.
Abu Abdallah Muhammad ibn Jabir al-Battani, who lived from 858 to 929, formally introduced the cosine function, as he built on the work of the Indians and Greeks. Muslim mathematicians also introduced the polar triangle for spherical triangles, sine and tangent tables created in 1/60th-of-a-degree steps. Nasir ad-Din at-Tusi, who lived from 1201 to 1274, wrote a book separating plane and spherical trigonometry into its own field of study, called the Book of the Transversal Figure. Muslim mathematicians revived the long-dead tangent function, invented by the Chinese but lost, and added the co-tangent, co-secant, and secant functions.
Georges Joachim defined trigonometric functions as ratios instead of lengths of lines during the 13th century. French mathematician François Viète, who lived from 1540 to 1653, introduced the polar triangle into spherical trigonometry and published two books, Canon Mathematicus, and Universalium Inspectionum Liber Singularis, in 1579. These two books were mathematical tables in which the values for sine are computed to 10 to the negative eighth power. In the 17th century, John Napier, a Scottish mathematician, invented logarithms, memory tricks to remember the 10 laws of how to solve spherical triangles; he also came up with what are now called Napier's analogies to help mathematicians solve oblique spherical triangles. In the 18th century, Leonhard Euler defined trigonometric functions in terms of complex numbers showing how basic laws of trigonometry were the consequences of arithmetic of complex numbers.
Early forms of trigonometry appeared in Chinese mathematics in the sixth century, but major advances in trigonometry did not happen until the 12th and 13th centuries, even though astronometrical calculations and calendar science demanded it. Shen Kuo, who lived from 1031 to 1095, used trigonometry to solve problems of chords and arcs. Guo Shoujing worked on arcs and circles, which formed the foundation of spherical trigonometry during the 12th and 13th centuries. Most of China's mathematics were lost after the Yuan Dynasty took root in 1271 until the 19th century.