# Trigonometry to the circle. The proof of the general

TrigonometryThe word of Trigonometry has been taken from the Greek, Trigonon,which means triangle and the other word, metron, has the meaning of measure.The trigonometry is the connecting link between mathematics and astronomy,between the way calendars are calculated, the Gnomon, and the sundial.

In theIslamic world, the calculation of the spherical triangles was necessary tocarry out the ritual custom. The direction to Mecca (qibla) wasindicated that the next to the hour lines on all public sundials.Muslim mathematicians have been contributed in this trigonometryfields. For a long time, the chord was used along with the sine. The magnitudetheory has been found in the al-Battani work around 858-929 CA.

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In hisastronomical work Islah al-Majisti (The Perfection of the Almagest), hesystematically employed the trigonometric function sine and versed sine witharguments between 0? and 180?. Since the cosine is defined as the sine of thecomplement of the angle and since no negative numbers are used, the versed sineis defined in the second quadrant as a sum of two quantities.The elements of trigonometry are set forth in an even moresystematic way in the Kitab al-Kamil (Perfect Book) of Abu I-Wafa(940-997/998). He defined several trigonometric functions in the circle withradius 1. The trigonometric tangent function is defined as a line on a tangentto the circle.The proof of the general spherical sine theorem was given by AbuI-Wafa in his al-Majisti (Almagest), by his pupil Abu Nasr ibn Iraq (d.

1036) in Risala fi ma’rifa al-qisi al-falakiyya (Treatise on theDetermination of Celestial Arcs), and by al-Khujandi, in the Kitab Maqalid’ilm al-hay’a (Book on the Keys of Astronomy).The use of trigonometry was expended through al-Biruni (973-1048).He is the author of the Mas ‘udic Canon, which is a summary of theresults from the works of many predecessors and of personal observations andcalculations. It comprises 11 books.

Book 3 is dedicated to trigonometry.Another important scholar in the area of trigonometry was Nasiral-Din al-Tusi (1201-1274). He wrote Kitab al-shakl al-qatta’ whichmeans The Book of the secant figure, also known as Treatise on the CompleteQuadrilateral. It was written in Persian and translated by the author intoArabic in 1260 possibly for the needs of the observatory of Maragha. While in thosefive books have contains a full system of trigonometric formulas for the planeand spherical triangles. This work played an important role in the developmentof mathematics in Europe. Algebra:            The word “Algebra”is derived from the Arabic language ?????(Al-Jabr) which means the completion or “reunion of brokenparts” and throve during the Islamic golden age (especially in the year830). Algebra comes from the treatise by the medieval Persian mathematician,Muhammad Ibn Musa Al-khwarizmi, who wrote a book with Arabic title, Kit?bal-mu?ta?ar f? ?is?b al-?abr wa-l-muq?bala, which can be translated as “TheCompendious Book on Calculation by Completion and Balancing”.

The treatiseprovided for the systematic solution of linear and quadratic equations. Theword al-jabr, presumably has meant restoration or completion and refer to thetransposition of subtracted terms to the other side of an equation. The word muqabalahis said to refer to reduction or balancing that is the cancellation of liketerms on opposite sides of the equation. The term is used by al-Khwarizmi todescribe the operations that he introduced, “reduction” and”balancing”, referring to the transposition of subtracted terms tothe other side of an equation, that is, the cancellation of like terms onopposite sides of the equation.The origin of Algebra can be traced back to ancient Babylon whichdeveloped a fairly complicated mathematical system, in which case they wereable to calculate in a similar way to algebra today.

Using this system, theyare able to apply formulas and compute solutions for unknown values ??forproblem classes that are usually solved by using Linear equations, Quadraticequations and Indefinite Linear Equations. In contrast, the Egyptians and mostof the Indian, Greek, and Chinese peoples in the first millennium, they are usuallystill using geometrical methods to solve such equations, such as thosementioned in “the Rhind Mathematical Papyrus”, “SulbaSutras”, “Eucilid’s Elements “And” The Nine Chapters on theMathematical Art “. The result of the Greeks in Geometry has been writtenin the book of elements, provides a framework for generalising the formula ofmathematics beyond the specific solution of a particular problem into the moregeneral system of expressing and solving equations, i.e. the framework of thededuction logic of thought.Besides Al-Khwarizmi, many other figures also have been developedin Algebra theorem.

Among others are:Al-Qalasadi: introduce algebraic symbols. The symbols weredeveloped in the 14th century by Ibnu al-Banna. Then, in the 15thcentury, Al-Qalasadi has been developed this. Al-Qalasadi has introduced themathematics symbols which use the characters of Arabic alphabet.Sharaf al-D?n al-Muzaffar ibn Muhammad ibn al-Muzaffar al-??s?(1135-1213) is a Mathematician and Islamic Astronomer from Persian. Sharifal-Din has been taught the various topics such mathematics, astronomy andrelated fields like numbers, astronomical tables, and astrology.

Al- ??s? hasbeen written several papers on Algebra. He gave the Ruffini-Horner methods to approach theroots of cubic equations. Although this method had previously been usedby Arab mathematicians to find almost the nth root of an integer, al- ??s? was the first to apply this method to solvethis general equation. In Al-Mu’adalat (About Equations), al-??s? findsalgebraic and numerical solutions of cubic equations and which first finds acubic polynomial derivative, an important result in differential calculus.

Omar Khayyam, a Persian scientist who has been developed thealgebra of geometry and found the general shape of the geometry of the cubic equation.