page_title: History of trigonometry
section_title: Medieval Islamic world
hierarchical_section_title: History of trigonometry / Development / Medieval Islamic world
caption_attribution_description: The Compendious Book on Calculation by Completion and Balancing: title page فارسی: صفحه‌ای از کتاب المختصر فی حساب الجبر والمقابله اثر خوارزمی
context_page_description: Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics and Babylonian mathematics.
  Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata, who discovered the sine function. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics and reaching its modern form with Leonhard Euler.
context_section_description: Previous works were later translated and expanded in the medieval Islamic world by Muslim mathematicians of mostly Persian and Arab descent, who enunciated a large number of theorems which freed the subject of trigonometry from dependence upon the complete quadrilateral, as was the case in Hellenistic mathematics due to the application of Menelaus' theorem. According to E. S. Kennedy, it was after this development in Islamic mathematics that "the first real trigonometry emerged, in the sense that only then did the object of study become the spherical or plane triangle, its sides and angles."
  Methods dealing with spherical triangles were also known, particularly the method of Menelaus of Alexandria, who developed "Menelaus' theorem" to deal with spherical problems. However, E. S. Kennedy points out that while it was possible in pre-Islamic mathematics to compute the magnitudes of a spherical figure, in principle, by use of the table of chords and Menelaus' theorem, the application of the theorem to spherical problems was very difficult in practice. In order to observe holy days on the Islamic calendar in which timings were determined by phases of the moon, astronomers initially used Menelaus' method to calculate the place of the moon and stars, though this method proved to be clumsy and difficult. It involved setting up two intersecting right triangles; by applying Menelaus' theorem it was possible to solve one of the six sides, but only if the other five sides were known. To tell the time from the sun's altitude, for instance, repeated applications of Menelaus' theorem were required. For medieval Islamic astronomers, there was an obvious challenge to find a simpler trigonometric method.
  In the early 9th century AD, Muhammad ibn Mūsā al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents. He was also a pioneer in spherical trigonometry. In 830 AD, Habash al-Hasib al-Marwazi produced the first table of cotangents. Muhammad ibn Jābir al-Harrānī al-Battānī (Albatenius) (853-929 AD) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.
  By the 10th century AD, in the work of Abū al-Wafā' al-Būzjānī, Muslim mathematicians were using all six trigonometric functions. Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values. He also developed the following trigonometric formula:
   (a special case of Ptolemy's angle-addition formula; see above)
  In his original text, Abū al-Wafā' states: "If we want that, we multiply the given sine by the cosine minutes, and the result is half the sine of the double". Abū al-Wafā also established the angle addition and difference identities presented with complete proofs:
  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".
  He also discovered the law of sines for spherical trigonometry:
  Also in the late 10th and early 11th centuries AD, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the following trigonometric identity:
  Al-Jayyani (989–1079) of al-Andalus wrote The book of unknown arcs of a sphere, which is considered "the first treatise on spherical trigonometry". It "contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle." This treatise later had a "strong influence on European mathematics", and his "definition of ratios as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influenced Regiomontanus.
  The method of triangulation was first developed by Muslim mathematicians, who applied it to practical uses such as surveying and Islamic geography, as described by Abu Rayhan Biruni in the early 11th century. Biruni himself introduced triangulation techniques to
caption: Page from The Compendious Book on Calculation by Completion and Balancing by Muhammad ibn Mūsā al-Khwārizmī (c. AD 820)
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