In a religious context, sin is an act of transgression against divine law. In Islamic ethics, Muslims see sin as anything that goes against the commands of Allah (God). Judaism regards the violation of any of the 613 commandments as a sin. In Jainism, sin refers to anything that harms the possibility of the jiva (being) to attain moksha (supreme emancipation).
The word derives from "Old English syn(n), for original *sunjō. The stem may be related to that of Latin 'sons, sont-is' guilty. In Old English there are examples of the original general sense, ‘offence, wrong-doing, misdeed'". The English Biblical terms translated as "sin" or "syn" from the Biblical Greek and Jewish terms sometimes originate from words in the latter languages denoting the act or state of missing the mark; the original sense of New Testament Greek ἁμαρτία hamartia "sin", is failure, being in error, missing the mark, especially in spear throwing; Hebrew hata "sin" originates in archery and literally refers to missing the "gold" at the centre of a target, but hitting the target, i.e. error. "To sin" has been defined from a Greek concordance as "to miss the mark".
In the Bahá'í Faith, humans are considered naturally good (perfect), fundamentally spiritual beings. Human beings were created because of God's immeasurable love. However, the Bahá'í teachings compare the human heart to a mirror, which, if turned away from the light of the sun (i.e. God), is incapable of receiving God's love.
There are a few differing Buddhist views on sin. American Zen author Brad Warner states that in Buddhism there is no concept of sin at all. The Buddha Dharma Education Association also expressly states "The idea of sin or original sin has no place in Buddhism."
In Buddhist thinking the whole universe, men as well as gods, are subject to a reign of law. Every action, good or bad, has an inevitable and automatic effect in a long chain of causes, an effect which is independent of the will of any deity. Even though this may leave no room for the concept of 'sin' in the sense of an act of defiance against the authority of a personal god, Buddhists speak of 'sin' when referring to transgressions against the universal moral code. (1974: 550)
However, Anantarika-kamma in Theravada Buddhism is a heinous crime, which through karmic process brings immediate disaster. In Mahayana Buddhism these five crimes are referred to as pañcānantarya (Pāli), and are mentioned in The Sutra Preached by the Buddha on the Total Extinction of the Dharma, The five crimes or sins are:
The doctrine of sin is central to Christianity, since its basic message is about redemption in Christ. Christian hamartiology describes sin as an act of offence against God by despising his persons and Christian biblical law, and by injuring others. In Christian views it is an evil human act, which violates the rational nature of man as well as God's nature and his eternal law. According to the classical definition of St. Augustine of Hippo sin is "a word, deed, or desire in opposition to the eternal law of God."
Among some scholars, sin is understood mostly as legal infraction or contract violation of non-binding philosophical frameworks and perspectives of Christian ethics, and so salvation tends to be viewed in legal terms.
Other Christian scholars understand sin to be fundamentally relational—a loss of love for the Christian God and an elevation of self-love ("concupiscence", in this sense), as was later propounded by Augustine in his debate with the Pelagians. As with the legal definition of sin, this definition also affects the understanding of Christian grace and salvation, which are thus viewed in relational terms.
Original sin, also called ancestral sin, is a Christian belief in the state of sin in which humanity has existed since the fall of man, stemming from Adam and Eve's rebellion in Eden, namely the sin of disobedience in consuming the forbidden fruit from the tree of the knowledge of good and evil.
This condition has been characterized in many ways, ranging from something as insignificant as a slight deficiency, or a tendency toward sin yet without collective guilt, referred to as a "sin nature", to something as drastic as total depravity or automatic guilt of all humans through collective guilt.
The concept of original sin was first alluded to in the 2nd century by Irenaeus, Bishop of Lyon in his controversy with certain dualist Gnostics. Other church fathers such as Augustine also shaped and developed the doctrine, seeing it as based on the New Testament teaching of Paul the Apostle (Romans 5:12–21 and 1 Corinthians 15:21-22) and the Old Testament verse of Psalms 51:5. Tertullian, Cyprian, Ambrose and Ambrosiaster considered that humanity shares in Adam's sin, transmitted by human generation. Augustine's formulation of original sin after 412 CE was popular among Protestant reformers, such as Martin Luther and John Calvin, who equated original sin with concupiscence (or "hurtful desire"), affirming that it persisted even after baptism and completely destroyed freedom to do good. Before 412 CE, Augustine said that free will was weakened but not destroyed by original sin. But after 412 CE this changed to a loss of free will except to sin. Modern Augustinian Calvinism holds this later view. The Jansenist movement, which the Catholic Church declared to be heretical, also maintained that original sin destroyed freedom of will. Instead the Catholic Church declares "Baptism, by imparting the life of Christ's grace, erases original sin and turns a man back towards God, but the consequences for nature, weakened and inclined to evil, persist in man and summon him to spiritual battle." "Weakened and diminished by Adam's fall, free will is yet not destroyed in the race."
In Hinduism, sin (Sanskrit: पाप pāpa "vice") describes actions that create negative karma by violating moral and ethical codes, which automatically brings negative consequences. This is somewhat similar to Abrahamic sin in the sense that pāpa is considered a crime against the laws of God, which is known as dharma, or moral order, and one's own self, but another term aparadha is used for grave offences.
However, the term papa cannot be taken in the literal sense as sin because there is no consensus regarding the nature of ultimate reality or God in Hinduism. Only, the Vedanta school being unambiguously theistic, whereas no anthropomorphic God exists in the rest of the five schools, Samkhya, Nyaya, Yoga, Vaisheshika, and Mīmāṃsā. The term papa however in the strictest sense refers to actions which bring about wrong/unfavourable consequences, not relating to a specific divine will in the absolute sense.
Sin is an important concept in Islamic ethics. Muslims see sin as anything that goes against the commands of Allah (God), a breach of the laws and norms laid down by religion. Islam teaches that sin is an act and not a state of being. It is believed that Allah weighs an individual’s good deeds against his or her sins on the Day of Judgement and punishes those individuals whose evil deeds outweigh their good deeds. These individuals are thought to be sentenced to afterlife in the fires of jahannum (Hell).
Islamic terms for sin include dhanb and khaṭīʾa, which are synonymous and refer to intentional sins; khiṭʾ, which means simply a sin; and ithm, which is used for grave sins.
A jiva (atman or soul) accumulates karma if it resorts to violence, non-chastity, falsehood, stealing, and possessiveness, if it hurts anyone, causes someone to hurt anyone, or commends hurting anyone by thought, speech or action. A jiva ceases to accumulate karma if it resorts to the ratnatraya (triple gems of Jainism): samyak gyan (right knowledge), samyak darshan (right sight) and samyak charitra (right character). A jiva begins to shed the accumulated karma by resorting to penance, repentance, vows and by exterminating foes of lust, anger, attachment, aversion, ignorance and fallacy.
No jiva can achieve moksha (release from samsara, the cycle of death and rebirth) without ceasing to accumulate karma and shedding the already accumulated karma entirely. Thus such a jiva is bound to remain in the worldly cycle of constant reincarnation, wherein it will keep taking rebirths, into any of the four broad types of living organisms, depending on the magnitude and nature of karma accumulated in previous birth(s). The four types are dev (beings of heaven, including deities), manushya (human), tiryanch (plants, animals, insects, etc.) and naarki (beings of hell).
During this cycle of getting born and dying for infinity, the jiva will have to then live the life of the organism he is and while living it, the jiva will again accumulate more karma. This will again lead to rebirth and again accumulating more karma. Thus, the cycle continues.
Jains believe that for complete liberation, not only the "sinful karma" but even the "meritorious karma" needs to be shed off. This means that a jiva can truly attain moksha only if the soul is completely and absolutely pure and devoid of any accumulation. For instance, sins may cause the jiva to be reborn in naraka (hell) and merits may cause it to be reborn in heaven. But heaven, like hell, is a part of worldly cycle of reincarnation and not supreme moksha of the soul. Thus, if a person hypothetically keeps performing only and exclusively good deeds in his life, he may still not attain moksha, because he has not yet shed off previously accumulated sins through repentance and knowledge.
Jains believe that only a human jiva has the capacity and the will to attain moksha. Hence the jiva should use this extremely rare opportunity of being born as a human to walk on the path that brings him closer to moksha. In fact, Jains take the concept of avoiding sin so seriously that not only are they completely vegetarian but some devout Jains also abstain from eating underground grown food like potatoes, onions, etc. to avoid killing small organisms. Most of the Jains are also nonalcoholics and eat before sunset each day.
Judaism regards the violation of any of the 613 commandments as a sin. Judaism teaches that to sin is a part of life, since there is no perfect man and everyone has an inclination to do evil "from his youth". Sin has many classifications and degrees. Some sins are punishable with death by the court, others with death by heaven, others with lashes, and others without such punishment, but no sins committed with willful intentions go without consequence. Sins committed out of lack of knowledge are not considered sins, since a sin can't be a sin if the one who did it didn't know it was wrong. Unintentional sins are considered less severe sins.
Sins between people are considered much more severe in Judaism than sins between man and God. Yom Kippur, the main day of repentance in Judaism, can atone for sins between man and God, but not for sins between man and his fellow, that is until he has appeased his friend. Eleazar ben Azariah derived [this from the verse]: "From all your sins before God you shall be cleansed" (Book of Leviticus, 16:30) – for sins between man and God Yom Kippur atones, but for sins between man and his fellow Yom Kippur does not atone until he appeases his fellow.
When the Temple yet stood in Jerusalem, people would offer Karbanot (sacrifices) for their misdeeds. The atoning aspect of karbanot is carefully circumscribed. For the most part, karbanot only expiate unintentional sins, that is, sins committed because a person forgot that this thing was a sin or by mistake. No atonement is needed for violations committed under duress or through lack of knowledge, and for the most part, karbanot cannot atone for a malicious, deliberate sin. In addition, karbanot have no expiating effect unless the person making the offering sincerely repents of his or her actions before making the offering, and makes restitution to any person who was harmed by the violation.
Judaism teaches that all willful sin has consequences. The completely righteous suffer for their sins (by humiliation, poverty, and suffering that God sends them) in this world and receive their reward in the world to come. The in-between (not completely righteous or completely wicked), suffer for and repent their sins after death and thereafter join the righteous. The very evil do not repent even at the gates of hell. Such people prosper in this world to receive their reward for any good deed, but cannot be cleansed by and hence cannot leave gehinnom, because they do not or cannot repent. This world can therefore seem unjust where the righteous suffer, while the wicked prosper. Many great thinkers have contemplated this.
They represented 'Adamu' as being tried before a god for the sin of casting down a divinity, a vestige of which is the 'original sin' which Christianity has tied up with Eve's disobedience in the Garden of Eden.
In astronomy, a celestial coordinate system (or celestial reference system) is a system for specifying positions of celestial objects: satellites, planets, stars, galaxies, and so on. Coordinate systems can specify an object's position in three-dimensional space or plot merely its direction on a celestial sphere, if the object's distance is unknown or trivial.
The coordinate systems are implemented in either spherical or rectangular coordinates. Spherical coordinates, projected on the celestial sphere, are analogous to the geographic coordinate system used on the surface of Earth. These differ in their choice of fundamental plane, which divides the celestial sphere into two equal hemispheres along a great circle. Rectangular coordinates, in appropriate units, are simply the cartesian equivalent of the spherical coordinates, with the same fundamental (x, y) plane and primary (x-axis) direction. Each coordinate system is named after its choice of fundamental plane.Chili con carne
Chili con carne or chilli con carne (/ˈtʃili koŋ ˈkaɾne/), meaning "chili with meat" and sometimes known as simply "chili" or "chilli", is a spicy stew containing chili peppers, meat (usually beef), and often tomatoes and beans. Other seasonings may include garlic, onions, and cumin. Geographic and personal tastes involve different types of meat and ingredients. Recipes provoke disputes among aficionados, some of whom insist that the word "chili" applies only to the basic dish, without beans and tomatoes. Chili con carne is a frequent dish for cook-offs and is used as an ingredient in other dishes.Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:
where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler's formula.
Euler's formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics".
When , Euler's formula evaluates to , which is known as Euler's identity.Final Fantasy X
Final Fantasy X is a role-playing video game developed and published by Square as the tenth entry in the Final Fantasy series. Originally released in 2001 for Sony's PlayStation 2, the game was re-released as Final Fantasy X/X-2 HD Remaster for PlayStation 3 and PlayStation Vita in 2013, for PlayStation 4 in 2015, Microsoft Windows in 2016, and for Nintendo Switch and Xbox One in 2019. The game marks the Final Fantasy series transition from entirely pre-rendered backdrops to fully three-dimensional areas, and is also the first in the series to feature voice acting. Final Fantasy X replaces the Active Time Battle (ATB) system with the "Conditional Turn-Based Battle" (CTB) system, and uses a new leveling system called the "Sphere Grid".
Set in the fantasy world of Spira, a setting influenced by the South Pacific, Thailand and Japan, the game's story revolves around a group of adventurers and their quest to defeat a rampaging monster known as Sin. The player character is Tidus, a star athlete in the fictional sport of blitzball, who finds himself in the world Spira after his home city of Zanarkand is destroyed by Sin. Shortly after arriving to Spira, Tidus joins the summoner Yuna on her pilgrimage to destroy Sin.
Development of Final Fantasy X began in 1999, with a budget of more than US$32.3 million (US$48.6 million in 2018 dollars) and a team of more than 100 people. The game was the first in the main series not entirely scored by Nobuo Uematsu; Masashi Hamauzu and Junya Nakano were signed as Uematsu's fellow composers. Final Fantasy X was both a critical and commercial success, selling over 8 million units worldwide on PlayStation 2. It is widely considered to be one of the greatest video games of all time. On March 13, 2003, it was followed by Final Fantasy X-2, making it the first Final Fantasy game to have a direct game sequel.Immaculate Conception
In Christian theology, the Immaculate Conception is the conception of the Virgin Mary free from original sin by virtue of the merits of her son Jesus. The Catholic Church teaches that God acted upon Mary in the first moment of her conception, keeping her "immaculate".The Immaculate Conception is commonly confused with the virgin birth of Jesus, the latter being, rather, the doctrine of the Incarnation. While many Christians believe in the virgin birth of Jesus, it is principally Roman Catholics, along with various other Christian denominations, who believe in the doctrine of the Immaculate Conception.
Although the belief that Mary was sinless, or conceived without original sin, has been widely held since Late Antiquity, the doctrine was not dogmatically defined in the Catholic Church until 1854 when Pope Pius IX, declared ex cathedra, i.e., using papal infallibility, in his papal bull Ineffabilis Deus, the Immaculate Conception to be doctrine. The Catholic Church celebrates the Solemnity of the Immaculate Conception on December 8; in many Catholic countries, it is a holy day of obligation or patronal feast, and in some a national public holiday.Latitude
In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle (defined below) which ranges from 0° at the Equator to 90° (North or South) at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the geodetic latitude as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular (or normal) to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six auxiliary latitudes which are used in special applications.List of trigonometric identities
In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.Original sin
Original sin, also called ancestral sin, is a Christian belief in the state of sin in which humanity has existed since the fall of man, stemming from Adam and Eve's rebellion in Eden, namely the sin of disobedience in consuming the forbidden fruit from the tree of the knowledge of good and evil. This condition has been characterized in many ways, ranging from something as insignificant as a slight deficiency, or a tendency toward sin yet without collective guilt, referred to as a "sin nature", to something as drastic as total depravity or automatic guilt of all humans through collective guilt.The concept of original sin was first alluded to in the 2nd century by Irenaeus, Bishop of Lyon in his controversy with certain dualist Gnostics. Other church fathers such as Augustine also shaped and developed the doctrine, seeing it as based on the New Testament teaching of Paul the Apostle (Romans 5:12–21 and 1 Corinthians 15:21-22) and the Old Testament verse of Psalms 51:5. Tertullian, Cyprian, Ambrose and Ambrosiaster considered that humanity shares in Adam's sin, transmitted by human generation. Augustine's formulation of original sin after 412 CE was popular among Protestant reformers, such as Martin Luther and John Calvin, who equated original sin with concupiscence (or "hurtful desire"), affirming that it persisted even after baptism and completely destroyed freedom to do good. Before 412 CE, Augustine said that free will was weakened but not destroyed by original sin. But after 412 CE this changed to a loss of free will except to sin. Modern Augustinian Calvinism holds this later view. The Jansenist movement, which the Catholic Church declared to be heretical, also maintained that original sin destroyed freedom of will. Instead the Catholic Church declares "Baptism, by imparting the life of Christ's grace, erases original sin and turns a man back towards God, but the consequences for nature, weakened and inclined to evil, persist in man and summon him to spiritual battle." "Weakened and diminished by Adam's fall, free will is yet not destroyed in the race."Refraction
In physics, refraction is the change in direction of a wave passing from one medium to another or from a gradual change in the medium. Refraction of light is the most commonly observed phenomenon, but other waves such as sound waves and water waves also experience refraction. How much a wave is refracted is determined by the change in wave speed and the initial direction of wave propagation relative to the direction of change in speed.
For light, refraction follows Snell's law, which states that, for a given pair of media, the ratio of the sines of the angle of incidence θ1 and angle of refraction θ2 is equal to the ratio of phase velocities (v1 / v2) in the two media, or equivalently, to the indices of refraction (n2 / n1) of the two media.
Optical prisms and lenses use refraction to redirect light, as does the human eye. The refractive index of materials varies with the wavelength of light, and thus the angle of the refraction also varies correspondingly. This is called dispersion and causes prisms and rainbows to divide white light into its constituent spectral colors.Robert Rodriguez
Robert Anthony Rodriguez (; born June 20, 1968) is an American filmmaker. He shoots, edits, produces, and scores many of his films in Mexico and his home state, Texas. Rodriguez directed the 1992 action film El Mariachi, which was a commercial success after grossing $2 million against a budget of $7,000. The film spawned two sequels known collectively as the Mexico Trilogy: Desperado and Once Upon a Time in Mexico. He directed From Dusk Till Dawn in 1996 and developed its television adaptation series (2014–2016). Rodriguez co-directed the 2005 neo-noir crime thriller anthology Sin City (adapted from the graphic novel of the same name) and the 2014 sequel, Sin City: A Dame to Kill For. Rodriguez also directed the Spy Kids films, The Faculty, The Adventures of Sharkboy and Lavagirl, Planet Terror, Machete, and Alita: Battle Angel. He is a friend and frequent collaborator of filmmaker Quentin Tarantino, who founded the production company A Band Apart, of which Rodriguez was a member. In December 2013, Rodriguez launched his own cable television channel, El Rey.Seven deadly sins
The seven deadly sins, also known as the capital vices, the seven traits of man, or cardinal sins, is a grouping and classification of vices within Christian teachings, although it does not appear explicitly in the Bible. Behaviours or habits are classified under this category if they directly give birth to other immoralities. According to the standard list, they are pride, greed, lust, envy, gluttony, wrath and sloth, which are also contrary to the seven heavenly virtues. These sins are often thought to be abuses or excessive versions of one's natural faculties or passions (for example, gluttony abuses one's desire to eat).
This classification originated with the desert fathers, especially Evagrius Ponticus, who identified seven or eight evil thoughts or spirits that one needed to overcome. Evagrius' pupil John Cassian, with his book The Institutes, brought the classification to Europe, where it became fundamental to Catholic confessional practices as evident in penitential manuals, sermons like "The Parson's Tale" from Chaucer's Canterbury Tales, and artworks like Dante's Purgatory (where the penitents of Mount Purgatory are depicted as being grouped and penanced according to the worst capital sin they committed). The Catholic Church used the concept of the deadly sins in order to help people curb their inclination towards evil before dire consequences and misdeeds could occur; the leader-teachers especially focused on pride (which is thought to be the sin that severs the soul from Grace, and the one that is representative and the very essence of all evil) and greed, both of which are seen as inherently sinful and as underlying all other sins to be prevented. To inspire people to focus on the seven deadly sins, the vices are discussed in treatises and depicted in paintings and sculpture decorations on Catholic churches as well as older textbooks.The seven deadly sins, along with the sins against the Holy Ghost and the sins that cry to Heaven for vengeance, are considered especially serious in the Western Christian traditions.Sin Cara
Jorge Arias (born September 5, 1977) is a Mexican-American professional wrestler currently signed to WWE, where he performs on the SmackDown brand under the ring name Sin Cara.Arias debuted on WWE’s main roster in 2011 as Sin Cara, temporarily replacing the original performer of the character Luis Urive. Arias subsequently began wrestling unmasked under the ring name Hunico, often in tag team competition alongside Camacho. In 2013, after the release of Urive, Arias reprised his role as Sin Cara and has since performed as the character. In contrast to Urive, Arias is the longest-tenured performer to portray Sin Cara, and his portrayal of the character is bilingual due to Arias having grown up in the United States, and thus speaks either Spanish or English, depending on the intended audience, whereas Urive legitimately does not know English. In 2014, Arias won the NXT Tag Team Championship alongside Kalisto as a part of The Lucha Dragons.
Before signing with WWE, Arias wrestled under the ring name Incognito or Mystico under which he had worked for professional wrestling promotions such as AAA in Mexico and Chikara in the United States.Sin City (film)
Sin City (a.k.a Frank Miller's Sin City) is a 2005 American neo-noir crime anthology film written, produced, directed by Robert Rodriguez and Frank Miller. Its based on Miller's graphic novel of the same name.Much of the film is based on the first, third, and fourth books in Miller's original comic series. The Hard Goodbye is about a man who embarks on a brutal rampage in search of his one-time sweetheart's killer, killing anyone, even the police, that gets in his way of finding and killing her murderer. The Big Fat Kill focuses on an everyman getting caught in a street war between a group of prostitutes and a group of mercenaries, the police and the mob. That Yellow Bastard follows an aging police officer who protects a young woman from a grotesquely disfigured serial killer. The intro and outro of the film are based on the short story "The Customer is Always Right" which is collected in Booze, Broads & Bullets, the sixth book in the comic series.
The film stars an ensemble cast led by Jessica Alba, Benicio del Toro, Brittany Murphy, Clive Owen, Mickey Rourke, Bruce Willis, and Elijah Wood. Featuring Alexis Bledel, Michael Clarke Duncan, Rosario Dawson, Carla Gugino, Rutger Hauer, Jaime King, Michael Madsen, Nick Stahl, and Makenzie Vega among others.
Sin City opened to wide critical and commercial success, gathering particular recognition for the film's unique color processing which rendered most of the film in black and white while retaining or adding color for selected objects. The film was screened at the 2005 Cannes Film Festival in competition and won the Technical Grand Prize for the film's "visual shaping".Sine
In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse).
More generally, the definition of sine (and other trigonometric functions) can be extended to any real value in terms of the length of a certain line segment in a unit circle. More modern definitions express the sine as an infinite series or as the solution of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.
The sine function is commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year.
The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin. The word "sine" (Latin "sinus") comes from a Latin mistranslation of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha.Spherical coordinate system
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.
The radial distance is also called the radius or radial coordinate. The polar angle may be called colatitude, zenith angle, normal angle, or inclination angle.
The use of symbols and the order of the coordinates differs between sources. In one system frequently encountered in physics (r, θ, φ) gives the radial distance, polar angle, and azimuthal angle, whereas in another system used in many mathematics books (r, θ, φ) gives the radial distance, azimuthal angle, and polar angle. In both systems ρ is often used instead of r. Other conventions are also used, so great care needs to be taken to check which one is being used.
A number of different spherical coordinate systems following other conventions are used outside mathematics. In a geographical coordinate system positions are measured in latitude, longitude and height or altitude. There are a number of different celestial coordinate systems based on different fundamental planes and with different terms for the various coordinates. The spherical coordinate systems used in mathematics normally use radians rather than degrees and measure the azimuthal angle counterclockwise from the x-axis to the y-axis rather than clockwise from north (0°) to east (+90°) like the horizontal coordinate system. The polar angle is often replaced by the elevation angle measured from the reference plane. Elevation angle of zero is at the horizon.
The spherical coordinate system generalizes the two-dimensional polar coordinate system. It can also be extended to higher-dimensional spaces and is then referred to as a hyperspherical coordinate system.Sun Yat-sen
Sun Yat-sen (; 12 November 1866 – 12 March 1925) was a Chinese politician, physician and philosopher who provisionally served as the first president of the Republic of China; and the first leader of the Kuomintang (Nationalist Party of China). He is referred as the "Father of the Nation" in the Republic of China due to his instrumental role in the overthrow of the Qing dynasty during the Xinhai Revolution. Sun remains a unique figure among 20th-century Chinese leaders for being widely revered in both mainland China and Taiwan.
Although Sun is considered to be one of the greatest leaders of modern China, his political life was one of constant struggle and frequent exile. After the success of the revolution and the Han Chinese regaining power after 268 years of living under Manchurian rule (Qing dynasty), he quickly resigned from his post as President of the newly founded Republic of China to Yuan Shikai, and led a successive revolutionary government as a challenge to the warlords who controlled much of the nation. Sun did not live to see his party consolidate its power over the country during the Northern Expedition. His party, which formed a fragile alliance with the Chinese Communist Party, split into two factions after his death.
Sun's chief legacy resides in his developing of the political philosophy known as the Three Principles of the People: nationalism (independence from foreign imperialist domination), "rights of the people" (sometimes translated as "democracy"), and the people's livelihood (just society).Triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted .
In Euclidean geometry any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted.Trigonometric functions
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 the simplest periodic functions, and as such are also widely used for studying periodic phenomena, through Fourier analysis.
The most familiar trigonometric functions are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used in modern mathematics.
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. For extending these definitions to functions whose domain is the whole projectively extended real line, one can use geometrical definitions using the standard unit circle (a circle with radius 1 unit). Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of the sine and the cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane from which some isolated points are removed.Trigonometry
Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. In particular, 3rd-century astronomers first noted that the ratio of the lengths of two sides of a right-angled triangle depends only on one acute angle of the triangle. These dependencies are now called trigonometric functions.
Trigonometry is the foundation of all applied geometry, including geodesy, surveying, celestial mechanics, solid mechanics, navigation.
Trigonometric functions have been extended as functions of a real or complex variable, which are today pervasive in all mathematics.
and describing the sins
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