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He is also credited with compiling the tables of sines and tangents at 15 ' intervals. He also introduced the secant and cosecant functions, as well studied the interrelations between the six trigonometric lines associated with an arc. His Almagest was widely read by medieval Arabic astronomers in the centuries after his death. He is known to have written several other books that have not survived.
Abu Al-Wafa' was the first to build a wall quadrant to observe the sky. It has been suggested that he was influenced by the works of Al-Battani as the latter describes a quadrant instrument in his Kit?b az-Z?j.
His use of tangent helped to solve problems involving right-angled spherical triangles, and developed a new technique to calculate sine tables, allowing him to construct more accurate tables than his predecessors.
In 997, he participated in an experiment to determine the difference in local time between his location, Baghdad, and that of al-Biruni (who was living in Kath, now a part of Uzbekistan). The result was very close to present-day calculations, showing a difference of approximately 1 hour between the two longitudes. Abu al-Wafa is also known to have worked with Ab? Sahl al-Q?h?, who was a famous maker of astronomical instruments. While what is extant from his works lacks theoretical innovation, his observational data were used by many later astronomers, including al-Biruni.
Among his works on astronomy, only the first seven treatises of his Almagest (Kit?b al-Majis) are now extant. The work covers numerous topics in the fields of plane and spherical trigonometry, planetary theory, and solutions to determine the direction of Qibla.
where A, B, C are the sides (measured in radians on the unit sphere) and a, b, c are the opposing angles.
Some sources suggest that he introduced the tangent function, although other sources give the credit for this innovation to al-Marwazi.
Almagest (? ?Kit?b al-Majis).
A book of zij called Z?j al-wi? (), no longer extant.
"A Book on Those Geometric Constructions Which Are Necessary for a Craftsman", (? ? ? Kit?b f? m? ya?t?j ilayh al-ni? min al-a?m?l al-handasiyya). This text contains over one hundred geometric constructions, including for a regular heptagon, which have been reviewed and compared with other mathematical treatises. The legacy of this text in Latin Europe is still debated.
"A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen", (? ? ? Kit?b f? m? ya?t?j ilayh al-kutt?b wa'l-?umm?l min ?ilm al-?is?b). This is the first book where negative numbers have been used in the medieval Islamic texts.