103-485: Abū al-Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī or Abū al-Wafā Būzhjānī ( Persian : ابو الوفا بوژگانی , Arabic : ابو الوفا بوزجانی ; 10 June 940 – 15 July 998) was a Persian mathematician and astronomer who worked in Baghdad . He made important innovations in spherical trigonometry , and his work on arithmetic for businessmen contains the first instance of using negative numbers in
206-562: A , b , c {\displaystyle 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 . He also wrote translations and commentaries on the algebraic works of Diophantus , al-Khwārizmī , and Euclid 's Elements . Persian language Russia Persian ( / ˈ p ɜːr ʒ ən , - ʃ ən / PUR -zhən, -shən ), also known by its endonym Farsi ( فارسی , Fārsī [fɒːɾˈsiː] ),
309-446: A medieval Islamic text. 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. He
412-638: A Dari dialect. In the 19th century, under the Qajar dynasty , the dialect that is spoken in Tehran rose to prominence. There was still substantial Arabic vocabulary, but many of these words have been integrated into Persian phonology and grammar. In addition, under the Qajar rule, numerous Russian , French , and English terms entered the Persian language, especially vocabulary related to technology. The first official attentions to
515-417: A corresponding inverse function , and an analog among the hyperbolic functions . The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles . To extend the sine and cosine functions to functions whose domain is the whole real line , geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then
618-418: A dictionary called Words of Scientific Association ( لغت انجمن علمی ), which was completed in the future and renamed Katouzian Dictionary ( فرهنگ کاتوزیان ). The first academy for the Persian language was founded on 20 May 1935, under the name Academy of Iran . It was established by the initiative of Reza Shah Pahlavi , and mainly by Hekmat e Shirazi and Mohammad Ali Foroughi , all prominent names in
721-500: A language name is first attested in English in the mid-16th century. Farsi , which is the Persian word for the Persian language, has also been used widely in English in recent decades, more often to refer to Iran's standard Persian. However, the name Persian is still more widely used. The Academy of Persian Language and Literature has maintained that the endonym Farsi is to be avoided in foreign languages, and that Persian
824-417: A literary language considerably different from the spoken Persian of the time. This became the basis of what is now known as "Contemporary Standard Persian". There are three standard varieties of modern Persian: All these three varieties are based on the classic Persian literature and its literary tradition. There are also several local dialects from Iran, Afghanistan and Tajikistan which slightly differ from
927-529: A period of several centuries, Ottoman Turkish (which was highly Persianised itself) had developed toward a fully accepted language of literature, and which was even able to lexically satisfy the demands of a scientific presentation. However, the number of Persian and Arabic loanwords contained in those works increased at times up to 88%. In the Ottoman Empire, Persian was used at the royal court, for diplomacy, poetry, historiographical works, literary works, and
1030-445: A rotation by an angle π {\displaystyle \pi } , the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of π {\displaystyle \pi } . That is, the equalities hold for any angle θ and any integer k . The algebraic expressions for the most important angles are as follows: Writing
1133-580: A speaker of Persian. Persian is a member of the Western Iranian group of the Iranian languages , which make up a branch of the Indo-European languages in their Indo-Iranian subdivision . The Western Iranian languages themselves are divided into two subgroups: Southwestern Iranian languages, of which Persian is the most widely spoken, and Northwestern Iranian languages, of which Kurdish and Balochi are
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#17330938836011236-595: A superscript after the symbol of the function denotes exponentiation , not function composition . For example sin 2 x {\displaystyle \sin ^{2}x} and sin 2 ( x ) {\displaystyle \sin ^{2}(x)} denote sin ( x ) ⋅ sin ( x ) , {\displaystyle \sin(x)\cdot \sin(x),} not sin ( sin x ) . {\displaystyle \sin(\sin x).} This differs from
1339-670: Is 360° (particularly in elementary mathematics ). However, in calculus and mathematical analysis , the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers , rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function , via power series, or as solutions to differential equations given particular initial values ( see below ), without reference to any geometric notions. The other four trigonometric functions ( tan , cot , sec , csc ) can be defined as quotients and reciprocals of sin and cos , except where zero occurs in
1442-605: Is a Western Iranian language belonging to the Iranian branch of the Indo-Iranian subdivision of the Indo-European languages . Persian is a pluricentric language predominantly spoken and used officially within Iran , Afghanistan , and Tajikistan in three mutually intelligible standard varieties , respectively Iranian Persian (officially known as Persian ), Dari Persian (officially known as Dari since 1964), and Tajiki Persian (officially known as Tajik since 1999). It
1545-644: Is a continuation of Middle Persian , an official language of the Sasanian Empire (224–651 CE), itself a continuation of Old Persian , which was used in the Achaemenid Empire (550–330 BCE). It originated in the region of Fars ( Persia ) in southwestern Iran. Its grammar is similar to that of many European languages. Throughout history, Persian was considered prestigious by various empires centered in West Asia , Central Asia , and South Asia . Old Persian
1648-610: Is a direct descendant of Middle and Old Persian. Gernot Windfuhr considers new Persian as an evolution of the Old Persian language and the Middle Persian language but also states that none of the known Middle Persian dialects is the direct predecessor of Modern Persian. Ludwig Paul states: "The language of the Shahnameh should be seen as one instance of continuous historical development from Middle to New Persian." The known history of
1751-812: Is also spoken natively in the Tajik variety by a significant population within Uzbekistan , as well as within other regions with a Persianate history in the cultural sphere of Greater Iran . It is written officially within Iran and Afghanistan in the Persian alphabet , a derivative of the Arabic script , and within Tajikistan in the Tajik alphabet , a derivative of the Cyrillic script . Modern Persian
1854-491: Is an arbitrary integer. Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence . Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets. More precisely, defining one has the following series expansions: The following continued fractions are valid in
1957-654: Is attested in Old Persian cuneiform on inscriptions from between the 6th and 4th century BC. Middle Persian is attested in Aramaic -derived scripts ( Pahlavi and Manichaean ) on inscriptions and in Zoroastrian and Manichaean scriptures from between the third to the tenth centuries (see Middle Persian literature ). New Persian literature was first recorded in the ninth century, after the Muslim conquest of Persia , since then adopting
2060-425: Is conventionally divided into three stages: Early New Persian remains largely intelligible to speakers of Contemporary Persian, as the morphology and, to a lesser extent, the lexicon of the language have remained relatively stable. New Persian texts written in the Arabic script first appear in the 9th-century. The language is a direct descendant of Middle Persian, the official, religious, and literary language of
2163-408: Is functions that are holomorphic in the whole complex plane, except some isolated points called poles . Here, the poles are the numbers of the form ( 2 k + 1 ) π 2 {\textstyle (2k+1){\frac {\pi }{2}}} for the tangent and the secant, or k π {\displaystyle k\pi } for the cotangent and the cosecant, where k
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#17330938836012266-455: Is not actually attested until 600 years later when it appears in the Sassanid era (224–651 AD) inscriptions, so any form of the language before this date cannot be described with any degree of certainty. Moreover, as a literary language, Middle Persian is not attested until much later, in the 6th or 7th century. From the 8th century onward, Middle Persian gradually began yielding to New Persian, with
2369-468: Is one of Afghanistan's two official languages, together with Pashto . The term Dari , meaning "of the court", originally referred to the variety of Persian used in the court of the Sasanian Empire in capital Ctesiphon , which was spread to the northeast of the empire and gradually replaced the former Iranian dialects of Parthia ( Parthian ). Tajik Persian ( форси́и тоҷикӣ́ , forsi-i tojikī ),
2472-570: Is one of the earliest attested Indo-European languages. According to certain historical assumptions about the early history and origin of ancient Persians in Southwestern Iran (where Achaemenids hailed from), Old Persian was originally spoken by a tribe called Parsuwash , who arrived in the Iranian Plateau early in the 1st millennium BCE and finally migrated down into the area of present-day Fārs province. Their language, Old Persian, became
2575-486: Is the appropriate designation of the language in English, as it has the longer tradition in western languages and better expresses the role of the language as a mark of cultural and national continuity. Iranian historian and linguist Ehsan Yarshater , founder of the Encyclopædia Iranica and Columbia University 's Center for Iranian Studies, mentions the same concern in an academic journal on Iranology , rejecting
2678-501: Is the unique solution with y (0) = 0 . The basic trigonometric functions can be defined by the following power series expansions . These series are also known as the Taylor series or Maclaurin series of these trigonometric functions: The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on
2781-403: Is then a definition of the real number π {\displaystyle \pi } which is independent of geometry. Applying the quotient rule to the tangent tan x = sin x / cos x {\displaystyle \tan x=\sin x/\cos x} , so the tangent function satisfies the ordinary differential equation It
2884-434: Is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that 2 π {\displaystyle 2\pi } is the smallest value for which they are periodic (i.e., 2 π {\displaystyle 2\pi } is the fundamental period of these functions). However, after
2987-454: Is used as its symbol in formulas. Today, the most common versions of these abbreviations are "sin" for sine, "cos" for cosine, "tan" or "tg" for tangent, "sec" for secant, "csc" or "cosec" for cosecant, and "cot" or "ctg" for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as
3090-486: The Encyclopædia Iranica notes that the Iranian, Afghan, and Tajiki varieties comprise distinct branches of the Persian language, and within each branch a wide variety of local dialects exist. The following are some languages closely related to Persian, or in some cases are considered dialects: More distantly related branches of the Iranian language family include Kurdish and Balochi . The Glottolog database proposes
3193-502: The Kalila wa Dimna . The language spread geographically from the 11th century on and was the medium through which, among others, Central Asian Turks became familiar with Islam and urban culture. New Persian was widely used as a trans-regional lingua franca , a task aided due to its relatively simple morphology, and this situation persisted until at least the 19th century. In the late Middle Ages, new Islamic literary languages were created on
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3296-438: The n th term lead to absolutely convergent series: Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions: The following infinite product for the sine is due to Leonhard Euler , and is of great importance in complex analysis: This may be obtained from the partial fraction decomposition of cot z {\displaystyle \cot z} given above, which
3399-651: The British colonization , Persian was widely used as a second language in the Indian subcontinent . It took prominence as the language of culture and education in several Muslim courts on the subcontinent and became the sole "official language" under the Mughal emperors . The Bengal Sultanate witnessed an influx of Persian scholars, lawyers, teachers, and clerics. Thousands of Persian books and manuscripts were published in Bengal. The period of
3502-576: The Buyid court. 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 described a quadrant instrument in his Kitāb az-Zīj . His use of the concept of the tangent helped solve problems involving right-angled spherical triangles . He 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
3605-459: The Euclidean plane that are related to the unit circle , which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between 0 and π 2 {\textstyle {\frac {\pi }{2}}} radians (90°), the unit circle definitions allow
3708-559: The Pythagorean identity . The other trigonometric functions can be found along the unit circle as By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is Since a rotation of an angle of ± 2 π {\displaystyle \pm 2\pi } does not change
3811-655: The Sultanate of Rum , Turkmen beyliks of Anatolia , Delhi Sultanate , the Shirvanshahs , Safavids , Afsharids , Zands , Qajars , Khanate of Bukhara , Khanate of Kokand , Emirate of Bukhara , Khanate of Khiva , Ottomans , and also many Mughal successors such as the Nizam of Hyderabad . Persian was the only non-European language known and used by Marco Polo at the Court of Kublai Khan and in his journeys through China. A branch of
3914-688: The Turkic , Armenian , Georgian , & Indo-Aryan languages . It also exerted some influence on Arabic, while borrowing a lot of vocabulary from it in the Middle Ages. Some of the world's most famous pieces of literature from the Middle Ages, such as the Shahnameh by Ferdowsi , the works of Rumi , the Rubáiyát of Omar Khayyám , the Panj Ganj of Nizami Ganjavi , The Divān of Hafez , The Conference of
4017-483: The ezāfe construction, expressed through ī (modern e/ye ), to indicate some of the relations between words that have been lost with the simplification of the earlier grammatical system. Although the "middle period" of the Iranian languages formally begins with the fall of the Achaemenid Empire, the transition from Old to Middle Persian had probably already begun before the 4th century BC. However, Middle Persian
4120-709: The function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation , for example sin( x ) . Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression sin x + y {\displaystyle \sin x+y} would typically be interpreted to mean sin ( x ) + y , {\displaystyle \sin(x)+y,} so parentheses are required to express sin ( x + y ) . {\displaystyle \sin(x+y).} A positive integer appearing as
4223-806: The reciprocal . For example sin − 1 x {\displaystyle \sin ^{-1}x} and sin − 1 ( x ) {\displaystyle \sin ^{-1}(x)} denote the inverse trigonometric function alternatively written arcsin x : {\displaystyle \arcsin x\colon } The equation θ = sin − 1 x {\displaystyle \theta =\sin ^{-1}x} implies sin θ = x , {\displaystyle \sin \theta =x,} not θ ⋅ sin x = 1. {\displaystyle \theta \cdot \sin x=1.} In this case,
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4326-585: The " Persianized " Turko-Mongol dynasties during the 12th to 15th centuries, and under restored Persian rule during the 16th to 19th centuries. Persian during this time served as lingua franca of Greater Persia and of much of the Indian subcontinent . It was also the official and cultural language of many Islamic dynasties, including the Samanids, Buyids , Tahirids , Ziyarids , the Mughal Empire , Timurids , Ghaznavids , Karakhanids , Seljuqs , Khwarazmians ,
4429-408: The (historically later) general functional notation in which f 2 ( x ) = ( f ∘ f ) ( x ) = f ( f ( x ) ) . {\displaystyle f^{2}(x)=(f\circ f)(x)=f(f(x)).} However, the exponent − 1 {\displaystyle {-1}} is commonly used to denote the inverse function , not
4532-672: The Birds by Attar of Nishapur , and the miscellanea of Gulistan and Bustan by Saadi Shirazi , are written in Persian. Some of the prominent modern Persian poets were Nima Yooshij , Ahmad Shamlou , Simin Behbahani , Sohrab Sepehri , Rahi Mo'ayyeri , Mehdi Akhavan-Sales , and Forugh Farrokhzad . There are approximately 130 million Persian speakers worldwide, including Persians , Lurs , Tajiks , Hazaras , Iranian Azeris , Iranian Kurds , Balochs , Tats , Afghan Pashtuns , and Aimaqs . The term Persophone might also be used to refer to
4635-558: The Middle Ages, and is because of the lack of the phoneme /p/ in Standard Arabic. The standard Persian of Iran has been called, apart from Persian and Farsi , by names such as Iranian Persian and Western Persian , exclusively. Officially, the official language of Iran is designated simply as Persian ( فارسی , fārsi ). The standard Persian of Afghanistan has been officially named Dari ( دری , dari ) since 1958. Also referred to as Afghan Persian in English, it
4738-532: The Ottoman Empire all spoke Persian, such as Sultan Selim I , despite being Safavid Iran's archrival and a staunch opposer of Shia Islam . It was a major literary language in the empire. Some of the noted earlier Persian works during the Ottoman rule are Idris Bidlisi 's Hasht Bihisht , which began in 1502 and covered the reign of the first eight Ottoman rulers, and the Salim-Namah , a glorification of Selim I. After
4841-521: The Persian language can be divided into the following three distinct periods: As a written language , Old Persian is attested in royal Achaemenid inscriptions. The oldest known text written in Old Persian is from the Behistun Inscription , dating to the time of King Darius I (reigned 522–486 BC). Examples of Old Persian have been found in what is now Iran , Romania ( Gherla ), Armenia , Bahrain , Iraq , Turkey, and Egypt . Old Persian
4944-425: The Persian language, a language historically called Dari, emerged in present-day Afghanistan. The first significant Persian poet was Rudaki . He flourished in the 10th century, when the Samanids were at the height of their power. His reputation as a court poet and as an accomplished musician and singer has survived, although little of his poetry has been preserved. Among his lost works are versified fables collected in
5047-401: The Persian model: Ottoman Turkish , Chagatai Turkic , Dobhashi Bengali , and Urdu, which are regarded as "structural daughter languages" of Persian. "Classical Persian" loosely refers to the standardized language of medieval Persia used in literature and poetry . This is the language of the 10th to 12th centuries, which continued to be used as literary language and lingua franca under
5150-848: The Perso-Arabic script. Persian was the first language to break through the monopoly of Arabic on writing in the Muslim world , with Persian poetry becoming a tradition in many eastern courts. It was used officially as a language of bureaucracy even by non-native speakers, such as the Ottomans in Anatolia , the Mughals in South Asia, and the Pashtuns in Afghanistan. It influenced languages spoken in neighboring regions and beyond, including other Iranian languages,
5253-479: The Sasanian Empire (224–651). However, it is not descended from the literary form of Middle Persian (known as pārsīk , commonly called Pahlavi), which was spoken by the people of Fars and used in Zoroastrian religious writings. Instead, it is descended from the dialect spoken by the court of the Sasanian capital Ctesiphon and the northeastern Iranian region of Khorasan , known as Dari. The region, which comprised
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#17330938836015356-575: The Sassanids (who were Persians, i.e. from the southwest) from the preceding Arsacids (who were Parthians, i.e. from the northeast). While Ibn al-Muqaffa' (eighth century) still distinguished between Pahlavi (i.e. Parthian) and Persian (in Arabic text: al-Farisiyah) (i.e. Middle Persian), this distinction is not evident in Arab commentaries written after that date. "New Persian" (also referred to as Modern Persian)
5459-457: The Seljuks, the Sultanate of Rum , took Persian language, art, and letters to Anatolia. They adopted the Persian language as the official language of the empire. The Ottomans , who can roughly be seen as their eventual successors, inherited this tradition. Persian was the official court language of the empire, and for some time, the official language of the empire. The educated and noble class of
5562-408: The angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete turn (360°) is an angle of 2 π (≈ 6.28) rad. For real number x , the notation sin x , cos x , etc. refers to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended,
5665-493: The area of Lake Urmia in the records of Shalmaneser III . The exact identity of the Parsuwash is not known for certain, but from a linguistic viewpoint the word matches Old Persian pārsa itself coming directly from the older word * pārćwa . Also, as Old Persian contains many words from another extinct Iranian language, Median , according to P. O. Skjærvø it is probable that Old Persian had already been spoken before
5768-457: The code fas for the dialects spoken across Iran and Afghanistan. This consists of the individual languages Dari ( prs ) and Iranian Persian ( pes ). It uses tgk for Tajik, separately. In general, the Iranian languages are known from three periods: namely Old, Middle, and New (Modern). These correspond to three historical eras of Iranian history ; Old era being sometime around the Achaemenid Empire (i.e., 400–300 BC), Middle era being
5871-512: The collapse of the Sassanid state, Parsik came to be applied exclusively to (either Middle or New) Persian that was written in the Arabic script . From about the 9th century onward, as Middle Persian was on the threshold of becoming New Persian, the older form of the language came to be erroneously called Pahlavi , which was actually but one of the writing systems used to render both Middle Persian as well as various other Middle Iranian languages. That writing system had previously been adopted by
5974-455: The definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number. Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry. Various ways exist in the literature for defining the trigonometric functions in a manner suitable for analysis; they include: Sine and cosine can be defined as
6077-458: The degree sign must be explicitly shown ( sin x° , cos x° , etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180 x / π )°, so that, for example, sin π = sin 180° when we take x = π . In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π /180 ≈ 0.0175. The six trigonometric functions can be defined as coordinate values of points on
6180-487: The denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle in radians. Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions. Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures. When radians (rad) are employed,
6283-612: 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 Abū Sahl al-Qūhī , 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
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#17330938836016386-474: The domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations . This allows extending the domain of sine and cosine functions to the whole complex plane , and the domain of the other trigonometric functions to the complex plane with some isolated points removed. Conventionally, an abbreviation of each trigonometric function's name
6489-498: The domain of trigonometric functions to be extended to all positive and negative real numbers. Let L {\displaystyle {\mathcal {L}}} be the ray obtained by rotating by an angle θ the positive half of the x -axis ( counterclockwise rotation for θ > 0 , {\displaystyle \theta >0,} and clockwise rotation for θ < 0 {\displaystyle \theta <0} ). This ray intersects
6592-404: The equivalent identities in terms of chords. The trigonometric identities he introduced were: He may have developed the law of sines for spherical triangles , though others like Abu-Mahmud Khojandi have been credited with the same achievement: where A , B , C {\displaystyle A,B,C} are the sides of the triangle (measured in radians on the unit sphere) and
6695-632: The extent of its influence on certain languages of the Indian subcontinent. Words borrowed from Persian are still quite commonly used in certain Indo-Aryan languages, especially Hindi - Urdu (also historically known as Hindustani ), Punjabi , Kashmiri , and Sindhi . There is also a small population of Zoroastrian Iranis in India, who migrated in the 19th century to escape religious execution in Qajar Iran and speak
6798-402: The first seven treatises of his Almagest ( Kitāb al-Majisṭī ) 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 . He defined the tangent function, and he established several trigonometric identities in their modern form, where the ancient Greek mathematicians had expressed
6901-485: The following phylogenetic classification: Tangent (trigonometry) In mathematics , the trigonometric functions (also called circular functions , angle functions or goniometric functions ) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry , such as navigation , solid mechanics , celestial mechanics , geodesy , and many others. They are among
7004-562: The formation of many modern languages in West Asia, Europe, Central Asia , and South Asia . Following the Turko-Persian Ghaznavid conquest of South Asia , Persian was firstly introduced in the region by Turkic Central Asians. The basis in general for the introduction of Persian language into the subcontinent was set, from its earliest days, by various Persianized Central Asian Turkic and Afghan dynasties. For five centuries prior to
7107-471: The formation of the Achaemenid Empire and was spoken during most of the first half of the first millennium BCE. Xenophon , a Greek general serving in some of the Persian expeditions, describes many aspects of Armenian village life and hospitality in around 401 BCE, which is when Old Persian was still spoken and extensively used. He relates that the Armenian people spoke a language that to his ear sounded like
7210-461: The language of the Persians. Related to Old Persian, but from a different branch of the Iranian language family, was Avestan , the language of the Zoroastrian liturgical texts. The complex grammatical conjugation and declension of Old Persian yielded to the structure of Middle Persian in which the dual number disappeared, leaving only singular and plural, as did gender. Middle Persian developed
7313-731: The line of equation y = 1 {\displaystyle y=1} at point C = ( x C , 1 ) . {\displaystyle \mathrm {C} =(x_{\mathrm {C} },1).} The tangent line to the unit circle at the point A , is perpendicular to L , {\displaystyle {\mathcal {L}},} and intersects the y - and x -axes at points D = ( 0 , y D ) {\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })} and E = ( x E , 0 ) . {\displaystyle \mathrm {E} =(x_{\mathrm {E} },0).} The coordinates of these points give
7416-432: The middle-period form only continuing in the texts of Zoroastrianism . Middle Persian is considered to be a later form of the same dialect as Old Persian. The native name of Middle Persian was Parsig or Parsik , after the name of the ethnic group of the southwest, that is, "of Pars ", Old Persian Parsa , New Persian Fars . This is the origin of the name Farsi as it is today used to signify New Persian. Following
7519-525: The most widely spoken. The term Persian is an English derivation of Latin Persiānus , the adjectival form of Persia , itself deriving from Greek Persís ( Περσίς ), a Hellenized form of Old Persian Pārsa ( 𐎱𐎠𐎼𐎿 ), which means " Persia " (a region in southwestern Iran, corresponding to modern-day Fars ). According to the Oxford English Dictionary , the term Persian as
7622-461: The nationalist movement of the time. The academy was a key institution in the struggle to re-build Iran as a nation-state after the collapse of the Qajar dynasty. During the 1930s and 1940s, the academy led massive campaigns to replace the many Arabic , Russian , French , and Greek loanwords whose widespread use in Persian during the centuries preceding the foundation of the Pahlavi dynasty had created
7725-495: The necessity of protecting the Persian language against foreign words, and to the standardization of Persian orthography , were under the reign of Naser ed Din Shah of the Qajar dynasty in 1871. After Naser ed Din Shah, Mozaffar ed Din Shah ordered the establishment of the first Persian association in 1903. This association officially declared that it used Persian and Arabic as acceptable sources for coining words. The ultimate goal
7828-414: The next period most officially around the Sasanian Empire , and New era being the period afterward down to present day. According to available documents, the Persian language is "the only Iranian language" for which close philological relationships between all of its three stages are established and so that Old, Middle, and New Persian represent one and the same language of Persian; that is, New Persian
7931-713: The northern part of Greece). Vardar Yenicesi differed from other localities in the Balkans insofar as that it was a town where Persian was also widely spoken. However, the Persian of Vardar Yenicesi and throughout the rest of the Ottoman-held Balkans was different from formal Persian both in accent and vocabulary. The difference was apparent to such a degree that the Ottomans referred to it as "Rumelian Persian" ( Rumili Farsisi ). As learned people such as students, scholars and literati often frequented Vardar Yenicesi, it soon became
8034-441: The numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values. Such simple expressions generally do not exist for other angles which are rational multiples of a right angle. The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees. G. H. Hardy noted in his 1908 work A Course of Pure Mathematics that
8137-529: The official language of the Achaemenid kings. Assyrian records, which in fact appear to provide the earliest evidence for ancient Iranian (Persian and Median) presence on the Iranian Plateau, give a good chronology but only an approximate geographical indication of what seem to be ancient Persians. In these records of the 9th century BCE, Parsuwash (along with Matai , presumably Medians) are first mentioned in
8240-417: The position or size of a shape, the points A , B , C , D , and E are the same for two angles whose difference is an integer multiple of 2 π {\displaystyle 2\pi } . Thus trigonometric functions are periodic functions with period 2 π {\displaystyle 2\pi } . That is, the equalities hold for any angle θ and any integer k . The same
8343-628: The present territories of northwestern Afghanistan as well as parts of Central Asia, played a leading role in the rise of New Persian. Khorasan, which was the homeland of the Parthians, was Persianized under the Sasanians. Dari Persian thus supplanted Parthian language , which by the end of the Sasanian era had fallen out of use. New Persian has incorporated many foreign words, including from eastern northern and northern Iranian languages such as Sogdian and especially Parthian. The transition to New Persian
8446-578: The region during the following centuries. Persian continued to act as a courtly language for various empires in Punjab through the early 19th century serving finally as the official state language of the Sikh Empire , preceding British conquest and the decline of Persian in South Asia. Beginning in 1843, though, English and Hindustani gradually replaced Persian in importance on the subcontinent. Evidence of Persian's historical influence there can be seen in
8549-589: The reign of Sultan Ghiyathuddin Azam Shah is described as the "golden age of Persian literature in Bengal". Its stature was illustrated by the Sultan's own correspondence and collaboration with the Persian poet Hafez ; a poem which can be found in the Divan of Hafez today. A Bengali dialect emerged among the common Bengali Muslim folk, based on a Persian model and known as Dobhashi ; meaning mixed language . Dobhashi Bengali
8652-434: The right-angled triangle definition, by taking the right-angled triangle to have the unit radius OA as hypotenuse . And since the equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} holds for all points P = ( x , y ) {\displaystyle \mathrm {P} =(x,y)} on the unit circle, this definition of cosine and sine also satisfies
8755-413: The same ordinary differential equation Sine is the unique solution with y (0) = 0 and y ′(0) = 1 ; cosine is the unique solution with y (0) = 1 and y ′(0) = 0 . One can then prove, as a theorem, that solutions cos , sin {\displaystyle \cos ,\sin } are periodic, having the same period. Writing this period as 2 π {\displaystyle 2\pi }
8858-401: The same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table. In geometric applications, the argument of a trigonometric function is generally the measure of an angle . For this purpose, any angular unit is convenient. One common unit is degrees , in which a right angle is 90° and a complete turn
8961-420: The simplest periodic functions , and as such are also widely used for studying periodic phenomena through Fourier analysis . The trigonometric functions most widely used in modern mathematics are the sine , the cosine , and the tangent functions. Their reciprocals are respectively the cosecant , the secant , and the cotangent functions, which are less used. Each of these six trigonometric functions has
9064-607: The site of a flourishing Persianate linguistic and literary culture. The 16th-century Ottoman Aşık Çelebi (died 1572), who hailed from Prizren in modern-day Kosovo , was galvanized by the abundant Persian-speaking and Persian-writing communities of Vardar Yenicesi, and he referred to the city as a "hotbed of Persian". Many Ottoman Persianists who established a career in the Ottoman capital of Constantinople (modern-day Istanbul ) pursued early Persian training in Saraybosna, amongst them Ahmed Sudi . The Persian language influenced
9167-561: The standard Persian of Tajikistan, has been officially designated as Tajik ( тоҷикӣ , tojikī ) since the time of the Soviet Union . It is the name given to the varieties of Persian spoken in Central Asia in general. The international language-encoding standard ISO 639-1 uses the code fa for the Persian language, as its coding system is mostly based on the native-language designations. The more detailed standard ISO 639-3 uses
9270-687: The standard Persian. The Hazaragi dialect (in Central Afghanistan and Pakistan), Herati (in Western Afghanistan), Darwazi (in Afghanistan and Tajikistan), Basseri (in Southern Iran), and the Tehrani accent (in Iran, the basis of standard Iranian Persian) are examples of these dialects. Persian-speaking peoples of Iran, Afghanistan, and Tajikistan can understand one another with a relatively high degree of mutual intelligibility . Nevertheless,
9373-457: The superscript could be considered as denoting a composed or iterated function , but negative superscripts other than − 1 {\displaystyle {-1}} are not in common use. If the acute angle θ is given, then any right triangles that have an angle of θ are similar to each other. This means that the ratio of any two side lengths depends only on θ . Thus these six ratios define six functions of θ , which are
9476-740: The trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, opposite represents the side opposite the given angle θ , and adjacent represents the side between the angle θ and the right angle. Various mnemonics can be used to remember these definitions. In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or π / 2 radians . Therefore sin ( θ ) {\displaystyle \sin(\theta )} and cos ( 90 ∘ − θ ) {\displaystyle \cos(90^{\circ }-\theta )} represent
9579-685: The unique solution to the initial value problem : Differentiating again, d 2 d x 2 sin x = d d x cos x = − sin x {\textstyle {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x} and d 2 d x 2 cos x = − d d x sin x = − cos x {\textstyle {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x} , so both sine and cosine are solutions of
9682-549: The unit circle at the point A = ( x A , y A ) . {\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).} The ray L , {\displaystyle {\mathcal {L}},} extended to a line if necessary, intersects the line of equation x = 1 {\displaystyle x=1} at point B = ( 1 , y B ) , {\displaystyle \mathrm {B} =(1,y_{\mathrm {B} }),} and
9785-478: The use of Farsi in foreign languages. Etymologically, the Persian term Farsi derives from its earlier form Pārsi ( Pārsik in Middle Persian ), which in turn comes from the same root as the English term Persian . In the same process, the Middle Persian toponym Pārs ("Persia") evolved into the modern name Fars. The phonemic shift from /p/ to /f/ is due to the influence of Arabic in
9888-418: The values of all trigonometric functions for any arbitrary real value of θ in the following manner. The trigonometric functions cos and sin are defined, respectively, as the x - and y -coordinate values of point A . That is, In the range 0 ≤ θ ≤ π / 2 {\displaystyle 0\leq \theta \leq \pi /2} , this definition coincides with
9991-415: The whole complex plane . Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation. Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions , that
10094-455: The whole complex plane: The last one was used in the historically first proof that π is irrational . There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match: This identity can be proved with the Herglotz trick. Combining the (– n ) th with
10197-578: Was already complete by the era of the three princely dynasties of Iranian origin, the Tahirid dynasty (820–872), Saffarid dynasty (860–903), and Samanid Empire (874–999). Abbas of Merv is mentioned as being the earliest minstrel to chant verse in the New Persian tongue and after him the poems of Hanzala Badghisi were among the most famous between the Persian-speakers of the time. The first poems of
10300-526: Was born in Buzhgan , (now Torbat-e Jam ) in Khorasan (in today's Iran). At age 19, in 959, he moved to Baghdad and remained there until his death in 998. He was a contemporary of the distinguished scientists Abū Sahl al-Qūhī and al-Sijzi who were in Baghdad at the time and others such as Abu Nasr Mansur , Abu-Mahmud Khojandi , Kushyar Gilani and al-Biruni . In Baghdad, he received patronage from members of
10403-586: Was patronised and given official status under the Sultans of Bengal , and was a popular literary form used by Bengalis during the pre-colonial period, irrespective of their religion. Following the defeat of the Hindu Shahi dynasty, classical Persian was established as a courtly language in the region during the late 10th century under Ghaznavid rule over the northwestern frontier of the subcontinent . Employed by Punjabis in literature, Persian achieved prominence in
10506-600: Was taught in state schools, and was also offered as an elective course or recommended for study in some madrasas . Persian learning was also widespread in the Ottoman-held Balkans ( Rumelia ), with a range of cities being famed for their long-standing traditions in the study of Persian and its classics, amongst them Saraybosna (modern Sarajevo , Bosnia and Herzegovina), Mostar (also in Bosnia and Herzegovina), and Vardar Yenicesi (or Yenice-i Vardar, now Giannitsa , in
10609-484: Was to prevent books from being printed with wrong use of words. According to the executive guarantee of this association, the government was responsible for wrongfully printed books. Words coined by this association, such as rāh-āhan ( راهآهن ) for "railway", were printed in Soltani Newspaper ; but the association was eventually closed due to inattention. A scientific association was founded in 1911, resulting in
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