Editing Law of sines (section)

==History==
An equivalent of the law of sines, that the sides of a triangle are proportional to the [[chord (trigonometry)|chords]] of double the opposite angles, was known to the 2nd century Hellenistic astronomer [[Ptolemy]] and used occasionally in his ''[[Almagest]]''.<ref>{{cite book |editor-last=Toomer |editor-first=Gerald J. |editor-link=Gerald J. Toomer |title=Ptolemy's Almagest |publisher=Princeton University Press |year=1998 |pages=[https://archive.org/details/ptolemys-almagest-toomer/page/7/mode/1up 7, fn. 10]; [https://archive.org/details/ptolemys-almagest-toomer/page/462/mode/1up 462, fn. 96] }}</ref>

Statements related to the law of sines appear in the astronomical and trigonometric work of 7th century Indian mathematician [[Brahmagupta]]. In his ''[[Brāhmasphuṭasiddhānta]]'', Brahmagupta expresses the circumradius of a triangle as the product of two sides divided by twice the [[Altitude (triangle)|altitude]]; the law of sines can be derived by alternately expressing the altitude as the sine of one or the other base angle times its opposite side, then equating the two resulting variants.<ref>{{cite book|last=Winter |first=Henry James Jacques |title=Eastern Science |publisher=[[John Murray (publishing house)|John Murray]] |year=1952 |page=46 |url=https://archive.org/details/easternscienceou0000wint/page/46/}} {{pb}} {{cite book|last=Colebrooke |first=Henry Thomas |title=Algebra, with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and Bhascara |location=London |publisher=[[John Murray (publishing house)|John Murray]] |year=1817 |pages=299–300 |url=https://archive.org/details/algebrawitharith00brahuoft/page/298/mode/2up}}</ref> An equation even closer to the modern law of sines appears in Brahmagupta's ''[[Khaṇḍakhādyaka]]'', in a method for finding the distance between the Earth and a planet following an [[epicycle]]; however, Brahmagupta never treated the law of sines as an independent subject or used it systematically for solving triangles.<ref>{{cite book |last=Van Brummelen |first=Glen |authorlink=Glen Van Brummelen |year=2009 |title=The Mathematics of the Heavens and the Earth |publisher=Princeton University Press |pages=109–111 |isbn=978-0-691-12973-0 }} {{pb}} {{cite book |last=Brahmagupta |title=The Khandakhadyaka: An Astronomical Treatise of Brahmagupta |translator=Sengupta |translator-first=Prabodh Chandra |year=1934 |publisher=University of Calcutta }}</ref>

The spherical law of sines is sometimes credited to 10th century scholars [[Abu-Mahmud Khujandi]] or [[Abū al-Wafāʾ]] (it appears in his ''Almagest''), but it is given prominence in [[Abū Naṣr Manṣūr]]'s ''Treatise on the Determination of Spherical Arcs'', and was credited to Abū Naṣr Manṣūr by his student [[al-Bīrūnī]] in his ''Keys to Astronomy''.<ref>{{cite book|last=Sesiano |first=Jacques |chapter=Islamic mathematics |pages=137–157 |title=Mathematics Across Cultures: The History of Non-western Mathematics|editor-first1=Helaine|editor-last1=Selin| editor-first2=Ubiratan|editor-last2=D'Ambrosio |year=2000 |publisher=Springer |isbn=1-4020-0260-2}} {{pb}} {{cite book |last=Van Brummelen |first=Glen |authorlink=Glen Van Brummelen |year=2009 |title=The Mathematics of the Heavens and the Earth |publisher=Princeton University Press |pages=183–185 |isbn=978-0-691-12973-0 }}</ref> [[Ibn Muʿādh al-Jayyānī]]'s 11th-century ''Book of Unknown Arcs of a Sphere'' also contains the spherical law of sines.<ref name="MacTutor Al-Jayyani">{{MacTutor|id=Al-Jayyani|title=Abu Abd Allah Muhammad ibn Muadh Al-Jayyani}}</ref> 

The 13th-century Persian mathematician [[Naṣīr al-Dīn al-Ṭūsī]] stated and proved the planar law of sines:<ref>{{Cite web |title=Nasir al-Din al-Tusi - Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Al-Tusi_Nasir/ |access-date=2025-03-10 |website=Maths History |language=en}}</ref> <blockquote>In any plane triangle, the ratio of the sides is equal to the ratio of the sines of the angles opposite to those sides. That is, in triangle ABC, we have AB : AC = Sin(∠ACB) : Sin(∠ABC)</blockquote> By employing the law of sines, al-Tusi could solve triangles where either two angles and a side were known or two sides and an angle opposite one of them were given. For triangles with two sides and the included angle, he divided them into right triangles that he could then solve. When three sides were given, he dropped a perpendicular line and then used Proposition II-13 of Euclid's ''Elements'' (a geometric version of the [[law of cosines]]). Al-Tusi established the important result that if the sum or difference of two arcs is provided along with the ratio of their sines, then the arcs can be calculated.<ref>{{Cite book |last=Katz |first=Victor J. |url=https://books.google.com/books?id=7rP2MAAACAAJ |title=A History of Mathematics: An Introduction |date=2017-03-21 |publisher=Pearson |isbn=978-0-13-468952-4 |pages=315 |language=en}}</ref>

According to [[Glen Van Brummelen]], "The Law of Sines is really [[Regiomontanus]]'s foundation for his solutions of right-angled triangles in Book IV, and these solutions are in turn the bases for his solutions of general triangles."<ref>{{cite book|first=Glen |last=Van Brummelen |year=2009 |url=https://books.google.com/books?id=bHD8IBaYN-oC |title=The Mathematics of the Heavens and the Earth: The Early History of Trigonometry |publisher=[[Princeton University Press]] |page=259 |isbn=978-0-691-12973-0}}</ref> Regiomontanus was a 15th-century German mathematician.