- published: 13 Aug 2007
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- author: khanacademy
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Clark University
Clark University is a private research university and liberal arts college in Worcester, Massachusetts. Founded in 1887, it is the oldest educational institution founded as an all-graduate university. Clark now also educates undergraduates. It is one of only three New England universities, along with Harvard and Yale, to be a founding member of the Association of American Universities, an organization of universities with the most prestigious profiles in research and graduate education. Clark withdrew its membership in 1999, citing a conflict with its mission; it is one of only four schools to do so. Clark is one of 40 schools profiled in the book Colleges That Change Lives by Loren Pope.
Clark University was founded by American businessman Jonas Gilman Clark in 1887. He started the university with a million dollars, and later added another million dollars to the university fund because he feared the university might someday face lack of funds. The university was opened on October 2, 1889 as the first all-graduate university in the United States.
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Germany
Germany (
i/ˈdʒɜrməni/), officially the Federal Republic of Germany (German: Bundesrepublik Deutschland, pronounced [ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant] (
listen)), is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate. With 81.8 million inhabitants, it is the most populous member state and the largest economy in the European Union. It is one of the major political powers of the European continent and a technological leader in many fields.
A region named Germania, inhabited by several Germanic peoples, was documented before AD 100. During the Migration Period, the Germanic tribes expanded southward, and established successor kingdoms throughout much of Europe. Beginning in the 10th century, German territories formed a central part of the Holy Roman Empire. During the 16th century, northern German regions became the centre of the Protestant Reformation while southern and western parts remained dominated by Roman Catholic denominations, with the two factions clashing in the Thirty Years' War, marking the beginning of the Catholic–Protestant divide that has characterized German society ever since. Occupied during the Napoleonic Wars, the rise of Pan-Germanism inside the German Confederation resulted in the unification of most of the German states into the German Empire in 1871 which was Prussian dominated. After the German Revolution of 1918–1919 and the subsequent military surrender in World War I, the Empire was replaced by the Weimar Republic in 1918, and partitioned in the Treaty of Versailles. Amidst the Great Depression, the Third Reich was proclaimed in 1933. The latter period was marked by Fascism and World War II. After 1945, Germany was divided by allied occupation, and evolved into two states, East Germany and West Germany. In 1990 Germany was reunified.
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Shape
The shape (Old English: gesceap, created thing) of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material composition.
Mathematician and statistician David George Kendall writes:
In this paper ‘shape’ is used in the vulgar sense, and means what one would normally expect it to mean. [...] We here define ‘shape’ informally as ‘all the geometrical information that remains when location, scale and rotational effects are filtered out from an object.’
Simple shapes can be described by basic geometry objects such as a set of two or more points, a line, a curve, a plane, a plane figure (e.g. square or circle), or a solid figure (e.g. cube or sphere). Most shapes occurring in the physical world are complex. Some, such as plant structures and coastlines, may be so arbitrary as to defy traditional mathematical description – in which case they may be analyzed by differential geometry, or as fractals.
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Archimedes
Archimedes of Syracuse (Ancient Greek: Ἀρχιμήδης; c. 287 BC – c. 212 BC) was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an explanation of the principle of the lever. He is credited with designing innovative machines, including siege engines and the screw pump that bears his name. Modern experiments have tested claims that Archimedes designed machines capable of lifting attacking ships out of the water and setting ships on fire using an array of mirrors.
Archimedes is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of pi. He also defined the spiral bearing his name, formulae for the volumes of surfaces of revolution and an ingenious system for expressing very large numbers.
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Aryabhata
Aryabhata (IAST: Āryabhaṭa, Sanskrit: आर्यभट) (476–550 CE) was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Āryabhaṭīya (499 CE, when he was 23 years old) and the Arya-siddhanta.
While there is a tendency to misspell his name as "Aryabhatta" by analogy with other names having the "bhatta" suffix, his name is properly spelled Aryabhata: every astronomical text spells his name thus, including Brahmagupta's references to him "in more than a hundred places by name". Furthermore, in most instances "Aryabhatta" does not fit the metre either.
Aryabhata mentions in the Aryabhatiya that it was composed 3,630 years into the Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476
Aryabhata was born in Taregna (literally, song of the stars), which is a small town in Bihar, India, about 30 km (19 mi) from Patna (then known as Pataliputra), the capital city of Bihar State. Evidences justify his birth there. In Taregna Aryabhata set up an Astronomical Observatory in the Sun Temple 6th century.
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Brook Taylor
Brook Taylor FRS (18 August 1685 – 29 December 1731) was an English mathematician who is best known for Taylor's theorem and the Taylor series.
Brook Taylor was born in Edmonton (at that time in Middlesex) to John Taylor of Bifrons House, Kent, and Olivia Tempest, daughter of Sir Nicholas Tempest, Bart., of Durham.[citation needed] Brook entered St John's College, Cambridge as a fellow-commoner in 1701, and took degrees of LL.B. and LL.D. in 1709 and 1714, respectively. Having studied mathematics under John Machin and John Keill, in 1708 he obtained a remarkable solution of the problem of the "centre of oscillation," which, however, remained unpublished until May 1714, when his claim to priority was disputed by Johann Bernoulli. Taylor's Methodus Incrementorum Directa et Inversa (1715) added a new branch to the higher mathematics, now designated the "calculus of finite differences." This work is available in translation on the web [1]. Among other ingenious applications, he used it to determine the form of movement of a vibrating string, by him first successfully reduced to mechanical principles. The same work contained the celebrated formula known as Taylor's theorem, the importance of which remained unrecognised until 1772, when J. L. Lagrange realized its powers and termed it "le principal fondement du calcul différentiel" ("the main foundation of differential calculus").
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Carl Benjamin Boyer
Carl Benjamin Boyer (November 3, 1906 – April 26, 1976) was a historian of sciences, and especially mathematics. David Foster Wallace called him the "Gibbon of math history". He wrote the books History of Analytic Geometry, The History of the Calculus and Its Conceptual Development, A History of Mathematics, and The Rainbow: From Myth to Mathematics. He served as book-review editor of Scripta Mathematica.
Boyer was valedictorian of his high school class. He received an A.B. from Columbia College in 1928 and an M.A. in 1929. He received his Ph.D. in Mathematics from Columbia University in 1939.
He was a full professor of Mathematics at Brooklyn College from 1952 until his death, although he had begun tutoring and teaching at Brooklyn College in 1928.
Boyer was instrumental as an inspiration to the founding of The Metropolitan New York Section of the History of Science Society.
He was a 1954 John Simon Guggenheim Memorial Foundation Fellow in History of Science & Technology.
He died of a heart attack in New York in 1976.
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Colin Maclaurin
Colin Maclaurin (February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a special case of the Taylor series, are named after him.
Owing to changes in orthography since that time (his name was originally rendered as e.g. "M'Laurine"), his surname is alternatively written MacLaurin. In Gaelic the name is "Cailean MacLabhruinn", which is literally 'Colin, the son of Laurence.'
Maclaurin was born in Kilmodan, Argyll. His father, Reverend and Minister of Glendaruel John Maclaurin, died when Maclaurin was in infancy, and his mother died before he reached nine years of age. He was then educated under the care of his uncle, the Reverend Daniel Maclaurin, minister of Kilfinnan.
At eleven, Maclaurin entered the University of Glasgow. He graduated MA three years later by defending a thesis on the Power of Gravity, and remained at Glasgow to study divinity until he was 19, when he was elected professor of mathematics in a ten-day competition at the Marischal College in the University of Aberdeen. He would hold the record as the world's youngest professor until March 2008, when the record was officially given to Alia Sabur.
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Eric W. Weisstein
Eric Wolfgang Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science (ScienceWorld). He currently works for Wolfram Research, Inc.
Weisstein holds a Ph.D. in planetary astronomy which he obtained from the California Institute of Technology's (Caltech) Division of Geological and Planetary Sciences in 1996 as well as an M.S. in planetary astronomy in 1993 also from Caltech. Weisstein graduated Cum Laude from Cornell University with a B.A. in physics and a minor in astronomy in 1990. During his summers away from Cornell, Weisstein participated in research at the Arecibo Observatory, a radio telescope facility in Puerto Rico operated by Cornell. As a graduate student, Weisstein also participated in research at Goddard Space Flight Center in Greenbelt, MD. During his time at Goddard, Weisstein participated in the development of hurricane visualization software. In 1996 Weisstein published his doctoral thesis titled Millimeter/Submillimeter Fourier Transform Spectroscopy of Jovian Planet Atmospheres which was completed under faculty advisor Dewey Muhleman and in association with Eugene Serabyn, who is now a member of the Caltech Jet Propulsion Laboratory (JPL).
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Euclid
Euclid (
/ˈjuːklɪd/ EWK-lid; Ancient Greek: Εὐκλείδης Eukleidēs), fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor.
"Euclid" is the anglicized version of the Greek name Εὐκλείδης, meaning "Good Glory".
Little is known about Euclid's life, as there are only a handful of references to him. The date and place of Euclid's birth and the date and circumstances of his death are unknown, and only roughly estimated in proximity to contemporary figures mentioned in references. No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art is the product of the artist's imagination.
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Gemma Frisius
Gemma Frisius (born Jemme Reinerszoon) (December 9, 1508–May 25, 1555), was a physician, mathematician, cartographer, philosopher, and instrument maker. He created important globes, improved the mathematical instruments of his day and applied mathematics in new ways to surveying and navigation.
Frisius was born in Dokkum, Friesland (present-day Netherlands) of poor parents, who died when he was young. He moved to Groningen and studied at the University in Leuven beginning in 1525. He received the degree of MD in 1536 and remained on the faculty of medicine in Leuven for the rest of his life. His oldest son, Cornelius Gemma, edited a posthumous volume of his work and continued to work with Ptolemaic astrological models.
While still a student, Frisius set up a workshop to produce globes and mathematical instruments. He became noted for the quality and accuracy of his instruments, which were praised by Tycho Brahe, among others. In 1533, he described for the first time the method of triangulation still used today in surveying. Twenty years later, he was the first to describe how an accurate clock could be used to determine longitude. Jean-Baptiste Morin (1583–1656) did not believe that Frisius' method for calculating longitude would work, remarking, "I do not know if the Devil will succeed in making a longitude timekeeper but it is folly for man to try."
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Hipparchus
Hipparchus/hɪˈpɑːrkəs/, or more correctly Hipparchos (Greek: Ἵππαρχος, Hipparkhos; c. 190 BC – c. 120 BC), was a Greek astronomer, geographer, and mathematician of the Hellenistic period. He is considered the founder of trigonometry.
Hipparchus was born in Nicaea (now Iznik, Turkey), and probably died on the island of Rhodes. He is known to have been a working astronomer at least from 162 to 127 BC. Hipparchus is considered the greatest ancient astronomical observer and, by some, the greatest overall astronomer of antiquity. He was the first whose quantitative and accurate models for the motion of the Sun and Moon survive. For this he certainly made use of the observations and perhaps the mathematical techniques accumulated over centuries by the Chaldeans from Babylonia. He developed trigonometry and constructed trigonometric tables, and he solved several problems of spherical trigonometry. With his solar and lunar theories and his trigonometry, he may have been the first to develop a reliable method to predict solar eclipses. His other reputed achievements include the discovery of Earth's precession, the compilation of the first comprehensive star catalog of the western world, and possibly the invention of the astrolabe, also of the armillary sphere, which he used during the creation of much of the star catalogue. It would be three centuries before Claudius Ptolemaeus' synthesis of astronomy would supersede the work of Hipparchus; it is heavily dependent on it in many areas.
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Leonhard Euler
Leonhard Euler (German pronunciation: [ˈɔʏlɐ], ( Swiss German pronunciation (help·info)) ( Standard German pronunciation (help·info)) English approximation, "Oiler"; 15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy.
Euler spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all."
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Nasir al-Din al-Tusi
Khawaja Muḥammad ibn Muḥammad ibn Ḥasan Ṭūsī (Persian: محمد بن محمد بن الحسن طوسی) (born 18 February 1201 in Ṭūs, Khorasan – died on 26 June 1274 in al-Kāżimiyyah district of metropolitan Baghdad), better known as Naṣīr al-Dīn al-Ṭūsī (Persian: نصیر الدین طوسی; or simply Tusi in the West), was a Persianpolymath and prolific writer: an astronomer, biologist, chemist, mathematician, philosopher, physician, physicist, scientist, theologian and Marja Taqleed. He was of the Ismaili-, and subsequently Twelver Shī‘ah Islamic belief. The Muslim scholar Ibn Khaldun (1332–1406) considered Tusi to be the greatest of the later Persian scholars.
Nasir al-Din Tusi was born in the city of Tus in medieval Khorasan (now in north-eastern Iran) in the year 1201 and began his studies at an early age. In Hamadan and Tus he studied the Qur'an, Hadith, Shi'a jurisprudence, logic, philosophy, mathematics, medicine and astronomy.
He was apparently born into a Shī‘ah family and lost his father at a young age. Fulfilling the wish of his father, the young Muhammad took learning and scholarship very seriously and travelled far and wide to attend the lectures of renowned scholars and acquire the knowledge which guides people to the happiness of the next world. At a young age he moved to Nishapur to study philosophy under Farid al-Din Damad and mathematics under Muhammad Hasib. He met also Farid al-Din al-'Attar, the legendary Sufi master who was later killed by Mongol invaders and attended the lectures of Qutb al-Din al-Misri.
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Nicolaus Copernicus
Nicolaus Copernicus (German: Nikolaus Kopernikus; Italian: Nicolò Copernico; Polish: Mikołaj Kopernik (help·info); in his youth, Niclas Koppernigk; 19 February 1473 – 24 May 1543) was a Renaissance astronomer and the first person to formulate a comprehensive heliocentric cosmology which displaced the Earth from the center of the universe.
Copernicus' epochal book, De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres), published just before his death in 1543, is often regarded as the starting point of modern astronomy and the defining epiphany that began the scientific revolution. His heliocentric model, with the Sun at the center of the universe, demonstrated that the observed motions of celestial objects can be explained without putting Earth at rest in the center of the universe. His work stimulated further scientific investigations, becoming a landmark in the history of science that is often referred to as the Copernican Revolution.
Among the great polymaths of the Renaissance, Copernicus was a mathematician, astronomer, jurist with a doctorate in law, physician, quadrilingual polyglot, classics scholar, translator, artist,Catholic cleric, governor, diplomat and economist.
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Ptolemy
Claudius Ptolemy ( /ˈtɒləmi/; Greek: Κλαύδιος Πτολεμαῖος, Klaudios Ptolemaios; Latin: Claudius Ptolemaeus; c. AD 90 – c. AD 168), was a Greek-Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the Thebaid. This theory, proposed by Theodore Meliteniotes, could be correct, but it is late (ca. 1360) and unsupported. There is no reason to suppose that he ever lived anywhere else than Alexandria, where he died around AD 168.
Ptolemy was the author of several scientific treatises, at least three of which were of continuing importance to later Islamic and European science. The first is the astronomical treatise now known as the Almagest (in Greek, Ἡ Μεγάλη Σύνταξις, "The Great Treatise", originally Μαθηματικὴ Σύνταξις, "Mathematical Treatise"). The second is the Geography, which is a thorough discussion of the geographic knowledge of the Greco-Roman world. The third is the astrological treatise known sometimes in Greek as the Apotelesmatika (Ἀποτελεσματικά), more commonly in Greek as the Tetrabiblos (Τετράβιβλος "Four books"), and in Latin as the Quadripartitum (or four books) in which he attempted to adapt horoscopic astrology to the Aristotelian natural philosophy of his day.
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Regiomontanus
Johannes Müller von Königsberg (6 June 1436 – 6 July 1476), today best known by his Latin toponym Regiomontanus, was a German mathematician, astronomer, astrologer, translator, instrument maker and Catholic bishop.
He was born in the Franconian village of Unfinden (now part of Königsberg, Bavaria) — not in the more famous East-Prussian Königsberg.
He was also known as Johannes der Königsberger (Johannes of Königsberg). His writings were published under the toponym Joannes de Monte Regio. The name Regiomontanus was first coined by Phillip Melanchthon in 1534, fifty-eight years after his death.
At eleven years of age, he became a student at the university in Leipzig, Saxony. In 1451 he continued his studies at Alma Mater Rudolfina, the university in Vienna, Austria. There he became a pupil and friend of Georg von Peurbach. In 1452 he graduated BA and was awarded his “magister artium” (Master of Arts) at the age of 21 in 1457 and held lectures in optics and ancient literature.
He continued to work with Peuerbach learning and extending the then known areas of astronomy, mathematics and instrument making until Peuerbach's death in 1461.
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karaoke
Chorus:
Brother's analyzing it
Other's strategizing it
416 or 905 who got the liver shit
Holocaust and homicidal shit rap scholar
Off the Walla, content heart Mic brawla
Brother's analyzing it
Other's strategizing it
416 or 905 who got the liver shit
Holocaust and homicidal shit rap scholar
Off the Walla, content heart mic brawla
Verse 1:
Peep the chemistry you niggaz couldn't fuck wit my legacy
Full court furry reign and microphone supremacy
First time my father breathed
Super star scholar seed Apollo creed notes'll make a brawler bleed
Rhyme record the microphone's a deadly murder weapon
Like Albe Shore's eyebrows shots be connecting
You ain't shit to me
Don't ever think of dissin' wit your sinistry
My ministry your wizardry writin' in calligraphy
Gothic my facial feature's like a prophet
You can't stop it I'm taking money out your pockets
I got crazy stamina far from an amateur
I'm representin Canada
Fuck you and your manager
I write smarter I strike harder
Brother's duck and cover up
Sorta like Clinton wit whitewater
Toronto golden child you know the dial
My flow is wild hitting Balboa style and never go to trial
The illa vet on the slicker set
Wit the bigger threat
I make a nigga flex
Fuck his bitch at the Cineplex
The rap champion from Brampton to the Hamptons
I keep the party ampin
Wit the mic temper tantrums
(Chorus)
Verse 2:
You pugilist or ludicrist
You either new to dis or taking buddah hits
I got the brain of 2 computer chips
(?) melly mel and dem
You'll never win, old school adrenaline
Verbal arsonist, larcenist slash narcissist
Rap styles nasty like a prostitute on jarvisist
Son I got the flyer brain
Rhymes so raw, I'm the brother who inspired Kane
Flow onto a higher plain
The beat maker, idol maker title taker
The fly flavor vocabulary skyscraper
Scam got the phater tracks
Tough like a battle cap
Props like word baccarat
Halifax and hacky sacks
Maestro's flavor's tight
I'm gonna reign tonight
Even Kanenites loves the way I flame the mic
Get the realist props at the meanest spots
Your girls on my jock you get tequila shots
Every time a needle drops
I be the rap savior from Asia to Malaysia
The 3 black kids from Yellowknife know I'm mega strife
Keep your eye on this rap scientist
Bitch niggaz still trying to do what I done in 89 and shit
(Chorus)
Verse 3:
Yo my dialogue brings attention
Sorta like the Shawshank redemption
I start adversaries wit the lyrics that I mention
I put a pro to shame niggaz know my name
Brother bow when they see me as if Jahova came
I got the betta shit forever slick
Peep the predicates
Bruce Banner grammar spray gamma ray
Don't make my sweater rip
I put a hoe in cheak
The microphone'll wreck
It's like a dime wit' no effect
Every time I blow connect
Fuck the calvelry I'm rolling wit the royal family
From Galilee to Bramely it's all about a salary
I eat you like a can o' Spam
I kick the framelan I slam a man
I ring more bells than Alexander Graham
Verbal Masicist who could flow as fast as this
Surpassing this fuck around your ass is hist'
It's a mystery
I be the first platinum act in history
I read the source article
Thanx for missing me
(Chorus)
Dean park, Dean park
My nigga Dime
My nigga Moses
[Maestro]
Maestro
This joint over here is dedicated to all the single mommys
All the single mommys worldwide this is for y'all
Maestro just hold your head
For real check this shit out
You're a black Mona Lisa named Tanisha
Bonafide features
Aires' my sign you're a libre
Got my open on your Etheopian attitude
Your body queen of the empire I feel your fire
Your eyes are hazel, and everytime that your perform graze my nasal
I wanna french kiss your naval
Got the joint I wanna put a lock on, box I wanna knock on
Give me a finger one time, I'll put a rock on
I used to dick many, I stick plenty
Now I sit back, relax, cool out
And meditate on a life I never thought of before
You got me open, look what's happenin'
Ill thoughts are vanishin'
I'm managin' to formulate goals and plans
Plus I know your daughter love me
'Cause everytime I leave your crib she wanna hug me
But don't sweat it, I'll treat her like my own
You're not alone, let's build a home
I'm in the zone
[Chorus: Jason Simmons]
We got it sewn, baby
We got it sewn
You're never alone
'Cause we got it sewn
[Maestro]
You make me wanna effervess when I hit your treasure chest
Never fess, finish college though, boo
Just one semester left
And they'll never be a cuter nurse, that's how I feel
It's me and you against the universe, you know the deal
I represent the land, you represent the water
Motivated by your strength, plus the way you raise your daughter
(on the real) She's a part of me, 'cause she's a party of you
To start a family, is what I wanna do
Alot of niggas can get a single mother stressed
Get the child attatched, then the brothers wanna jet
You know my repitoire, I'll never leave you
'Cause you're top-notch
Just like the cutie that my pops got
I hold it down with your crib still
I taught your daughter how to read and plus I'll help you keep your fridge
fill
So study hard, I got your back, you're not alone
Let's build a home, I'm in the zone
[Chorus]
[Jason Simmons]
I'll be the shoulder you can cry on
I'll always be around
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All the single mommys, just hold your head
Mr. Maes' signing out now
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Basic Trigonometry
- Order: Reorder
- Duration: 8:44
- Updated: 17 Dec 2012
An introduction to trigonometric functions: sine, cosine, and tangent.
http://wn.com/Basic_Trigonometry
An introduction to trigonometric functions: sine, cosine, and tangent.
Trigonometry - The Trig Ratios, sin x, cos x and tan x : ExamSolutions
- Order: Reorder
- Duration: 7:39
- Updated: 09 Dec 2012
In this tutorial you are shown how the sin, cos and tan ratios compare two sides of a right angled triangle.. You need to know this before calculating lengths and angles of right angled triangles. To see this video and other go to ExamSolutions at www.examsolutions.co.uk
http://wn.com/Trigonometry_The_Trig_Ratios,_sin_x,_cos_x_and_tan_x_ExamSolutions
In this tutorial you are shown how the sin, cos and tan ratios compare two sides of a right angled triangle.. You need to know this before calculating lengths and angles of right angled triangles. To see this video and other go to ExamSolutions at www.examsolutions.co.uk
- published: 07 Sep 2009
- views: 18099
- author: ExamSolutions
Sin Cos Tan - Basic Trigonometry - YourTeacher.com
- Order: Reorder
- Duration: 3:33
- Updated: 16 Dec 2012
For a complete lesson on sin cos tan, go to www.yourteacher.com - 1000+ online math lessons featuring a personal math teacher inside every lesson! In this lesson, students learn to find the missing side lengths and the missing angle measures in right triangles using sine, cosine, and tangent. Note that a scientific or graphing calculator is required for the problems in this lesson.
http://wn.com/Sin_Cos_Tan_Basic_Trigonometry_YourTeachercom
For a complete lesson on sin cos tan, go to www.yourteacher.com - 1000+ online math lessons featuring a personal math teacher inside every lesson! In this lesson, students learn to find the missing side lengths and the missing angle measures in right triangles using sine, cosine, and tangent. Note that a scientific or graphing calculator is required for the problems in this lesson.
- published: 31 Oct 2007
- views: 461201
- author: yourteachermathhelp
Basic Trigonometry Formula - SOH-CAH-TOA
- Order: Reorder
- Duration: 5:16
- Updated: 09 Dec 2012
www.FreedomUniversity.TV. A basic introduction to defining the three major trigonmetric functions (sine, cosine, tangent) are discussed using the mneumonic SOH-CAH-TOA. For more applications, examples and other topics please contact Dr J at john@e-liteworks.com or visit http
http://wn.com/Basic_Trigonometry_Formula_SOH-CAH-TOA
www.FreedomUniversity.TV. A basic introduction to defining the three major trigonmetric functions (sine, cosine, tangent) are discussed using the mneumonic SOH-CAH-TOA. For more applications, examples and other topics please contact Dr J at john@e-liteworks.com or visit http
- published: 01 Jul 2008
- views: 138340
- author: mathworksheet
Introduction to Trigonometric Functions
- Order: Reorder
- Duration: 9:33
- Updated: 09 Dec 2012
A brief lesson on trigonometric functions: sine, cosine, and tangent. We look at their definition and an application.
http://wn.com/Introduction_to_Trigonometric_Functions
A brief lesson on trigonometric functions: sine, cosine, and tangent. We look at their definition and an application.
- published: 03 Jan 2010
- views: 70418
- author: outofmylaboratory
Math - Trigonometry - How to use Sin, Cos or Tan to find the length of a side.
- Order: Reorder
- Duration: 14:36
- Updated: 16 Dec 2012
Expertmathstutor how to use trigonometry to find the length of a side using Sin, Cos and Tan with right angled triangles.
http://wn.com/Math_Trigonometry_How_to_use_Sin,_Cos_or_Tan_to_find_the_length_of_a_side
Expertmathstutor how to use trigonometry to find the length of a side using Sin, Cos and Tan with right angled triangles.
- published: 27 Nov 2010
- views: 18987
- author: expertmathstutor
Basic Trigonometry
- Order: Reorder
- Duration: 9:17
- Updated: 16 Dec 2012
Learn more: www.khanacademy.org Introduction to trigonometry
http://wn.com/Basic_Trigonometry
Learn more: www.khanacademy.org Introduction to trigonometry
- published: 15 Feb 2011
- views: 475559
- author: khanacademy
Solving Trigonometric Equations
- Order: Reorder
- Duration: 10:06
- Updated: 16 Dec 2012
Solving Trigonometric Equations! That is, solving equations with trig in them! For more free math videos, visit PatrickJMT.com Austin Math Tutor, Austin Math Tutoring, Austin Algebra Tutor, Austin Calculus Tutor
http://wn.com/Solving_Trigonometric_Equations
Solving Trigonometric Equations! That is, solving equations with trig in them! For more free math videos, visit PatrickJMT.com Austin Math Tutor, Austin Math Tutoring, Austin Algebra Tutor, Austin Calculus Tutor
- published: 17 Mar 2008
- views: 163438
- author: patrickJMT
Calculus I - Lecture 03 - Trigonometry for Calculus
- Order: Reorder
- Duration: 72:41
- Updated: 17 Dec 2012
This video covers the following topics (times listed are approximate) Right Triangle Trigonometry [11.5 min.] Trigonometric Graphs [5.5 min.] Handy Trigonometric Identities [5.5 min.] Laws of Sine and Cosine [4 min.] Trigonometric Families [4 min.]
http://wn.com/Calculus_I_Lecture_03_Trigonometry_for_Calculus
This video covers the following topics (times listed are approximate) Right Triangle Trigonometry [11.5 min.] Trigonometric Graphs [5.5 min.] Handy Trigonometric Identities [5.5 min.] Laws of Sine and Cosine [4 min.] Trigonometric Families [4 min.]
- published: 11 Nov 2009
- views: 45653
- author: UMKC
Introduction to Trigonometric Functions Using Triangles
- Order: Reorder
- Duration: 9:38
- Updated: 26 Nov 2012
Defines trigonometric functions using right triangles. mathispower4u.wordpress.com
http://wn.com/Introduction_to_Trigonometric_Functions_Using_Triangles
Defines trigonometric functions using right triangles. mathispower4u.wordpress.com
- published: 26 Feb 2010
- views: 5714
- author: bullcleo1
Trigonometric Ratios - SohCahToa
- Order: Reorder
- Duration: 6:04
- Updated: 15 Dec 2012
This is a video tutorial on the Trigonometric Ratios, Sine, Cosine and Tangent. This video tutorial will help you remember the trig ratios using the acronym, sohcatoa. At the end of the video there will be some sample problems. If you have requests, suggestions or ideas for the show, please contact me at mr.alvarez.math@gmail.com
http://wn.com/Trigonometric_Ratios_SohCahToa
This is a video tutorial on the Trigonometric Ratios, Sine, Cosine and Tangent. This video tutorial will help you remember the trig ratios using the acronym, sohcatoa. At the end of the video there will be some sample problems. If you have requests, suggestions or ideas for the show, please contact me at mr.alvarez.math@gmail.com
- published: 15 Feb 2011
- views: 37977
- author: despicablemath
Deriving Trigonometric Identies from Known Identities
- Order: Reorder
- Duration: 9:16
- Updated: 11 Dec 2012
Deriving Trigonometric Identies from Known Identities - In this video I show that if you know about 2 or 3 identities and a few definitions, you can actually derive lots of other trigonometric identities easily! For more free math videos, visit JustMathTutoring.com
http://wn.com/Deriving_Trigonometric_Identies_from_Known_Identities
Deriving Trigonometric Identies from Known Identities - In this video I show that if you know about 2 or 3 identities and a few definitions, you can actually derive lots of other trigonometric identities easily! For more free math videos, visit JustMathTutoring.com
- published: 07 Sep 2008
- views: 41935
- author: patrickJMT
The unit circle definition of trigonometric function
- Order: Reorder
- Duration: 9:07
- Updated: 15 Dec 2012
Learn more: www.khanacademy.org Using the unit circle to extend the SOH CAH TOA definition of the basic trigonometric functions.
http://wn.com/The_unit_circle_definition_of_trigonometric_function
Learn more: www.khanacademy.org Using the unit circle to extend the SOH CAH TOA definition of the basic trigonometric functions.
- published: 22 Aug 2007
- views: 358215
- author: khanacademy
Tree trigonometry
- Order: Reorder
- Duration: 1:48
- Updated: 06 Dec 2012
Markus Eichhorn shows off a clever trick for estimating the height of trees! Markus is a tree expert and lecturer in ecology at the University of Nottingham. More about him and Nottingham scientists at www.test-tube.org.uk
http://wn.com/Tree_trigonometry
Markus Eichhorn shows off a clever trick for estimating the height of trees! Markus is a tree expert and lecturer in ecology at the University of Nottingham. More about him and Nottingham scientists at www.test-tube.org.uk
- published: 07 May 2008
- views: 11764
- author: nottinghamscience
Oh No, Pi Politics Again
- Order: Reorder
- Duration: 3:39
- Updated: 17 Dec 2012
Hooray, Michael Blake's video is now up here: youtu.be I've got opinions. Facts, too.
http://wn.com/Oh_No,_Pi_Politics_Again
Hooray, Michael Blake's video is now up here: youtu.be I've got opinions. Facts, too.
- published: 31 Mar 2011
- views: 439062
- author: Vihart
Trig: Solving Equations 1
- Order: Reorder
- Duration: 7:53
- Updated: 17 Dec 2012
Introduction to solving trigonometric equations.
http://wn.com/Trig_Solving_Equations_1
Introduction to solving trigonometric equations.
- published: 09 Nov 2007
- views: 131735
- author: MathTV
Trigonometry: Identities
- Order: Reorder
- Duration: 4:42
- Updated: 15 Dec 2012
More trig videos at www.mathtv.com.
http://wn.com/Trigonometry_Identities
Solving Trigonometric Equations - Easy
- Order: Reorder
- Duration: 5:40
- Updated: 14 Dec 2012
A short worked example: Solving a trigonometric equation. Please visit my site "animatingmath.com" where videos are all presented by topic area. It's all free, I'm trying to promote it.
http://wn.com/Solving_Trigonometric_Equations_Easy
A short worked example: Solving a trigonometric equation. Please visit my site "animatingmath.com" where videos are all presented by topic area. It's all free, I'm trying to promote it.
- published: 14 Mar 2008
- views: 37618
- author: dcollis2
Right Triangle Trigonometry
- Order: Reorder
- Duration: 3:13
- Updated: 13 Dec 2012
This video explores basic right triangle trigonometry used to locate a missing side in a right triangle.
http://wn.com/Right_Triangle_Trigonometry
This video explores basic right triangle trigonometry used to locate a missing side in a right triangle.
- published: 21 Jan 2008
- views: 45791
- author: kthomp271828
Trigonometry Tutorial 1 - What is Trigonometry?
- Order: Reorder
- Duration: 5:32
- Updated: 11 Dec 2012
Here we learn what Trigonometry is and learn what the six trigonometric functions are. Visit my website at thebosscoder.webs.com!
http://wn.com/Trigonometry_Tutorial_1_What_is_Trigonometry?
Here we learn what Trigonometry is and learn what the six trigonometric functions are. Visit my website at thebosscoder.webs.com!
- published: 31 Dec 2011
- views: 1653
- author: TheBossCoder
Trigonometry: The Law of Cosines
- Order: Reorder
- Duration: 6:21
- Updated: 16 Dec 2012
More at www.MathTV.com
http://wn.com/Trigonometry_The_Law_of_Cosines
WildTrig4: Five main laws of rational trigonometry
- Order: Reorder
- Duration: 9:48
- Updated: 09 Dec 2012
We derive from first principles the main laws of rational trigonometry, using the concepts of quadrance and spread to replace the usual distance and angle. Most everything works out much simpler.
http://wn.com/WildTrig4_Five_main_laws_of_rational_trigonometry
We derive from first principles the main laws of rational trigonometry, using the concepts of quadrance and spread to replace the usual distance and angle. Most everything works out much simpler.
- published: 12 Nov 2007
- views: 12453
- author: njwildberger
Trigonometric Substitution - Example 1
- Order: Reorder
- Duration: 14:09
- Updated: 16 Dec 2012
Trigonometric Substitution - Example 1. Just a basic trigonometric substitution problem (still long though!). I show the basic substitutions along with how to use the right triangle to get back to the original variable.
http://wn.com/Trigonometric_Substitution_Example_1
Trigonometric Substitution - Example 1. Just a basic trigonometric substitution problem (still long though!). I show the basic substitutions along with how to use the right triangle to get back to the original variable.
- published: 07 Feb 2011
- views: 70843
- author: patrickJMT
photo: AP / Jason DeCrow
16 Dec 2012
NEWTOWN, Conn. — The gunman in the Connecticut shooting rampage committed suicide as first responders closed in, the governor said Sunday, raising the specter that Adam Lanza had planned an even more gruesome massacre and was stopped short. Lanza blasted his way into the building and used a high-power rifle to kill 20 children and six adults, including the principal who tried to stop him, authorities said ... Among the questions ... H ... The Rev....(size: 8.8Kb)
photo: AP / Guang Niu, Pool
16 Dec 2012
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photo: AP / David Goldman
17 Dec 2012
WASHINGTON—Democrats say meaningful action in the wake of last week's elementary school shooting must include a ban on military-style assault weapons and a look at how the nation deals with individuals suffering from serious mental illness. Several Democratic lawmakers and Independent Sen ... Gun control was a hot topic in the early 1990s, when Congress enacted a 10-year ban on assault weapons ... He added ... Sen ... Rep ... ———....(size: 5.3Kb)
photo: AP / Newtown Bee, Shannon Hicks
15 Dec 2012
Newtown, Conn. Twenty schoolchildren were slaughtered by a heavily armed gunman who opened fire at a suburban elementary school in Connecticut on Friday, ultimately killing at least 27 people including himself in the one of the worst mass shootings in US history ... Authorities found 18 children and seven adults, including the gunman, dead at the school, and two children were pronounced dead later after being taken to a hospital ... Reuters ....(size: 7.5Kb)
photo: AP / Jessica Hill
14 Dec 2012
More than two dozen people, mostly elementary school children, were shot and killed at a Newtown, Conn., elementary school this morning, federal and state sources tell ABC News. The massacre involved two gunmen at a Connecticut school this morning, prompting the town of Newtown to lock down all of its schools and draw SWAT teams to the school, authorities said today. State Police confirm that one shooter is dead ... at ABCNews.com ... ....(size: 4.5Kb)
The Guardian
14 Dec 2012
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12 Dec 2012
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Sacramento Bee
11 Dec 2012
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11 Dec 2012
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The Daily Telegraph
11 Dec 2012
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10 Dec 2012
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Japan Times
07 Dec 2012
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Irish Times
30 Nov 2012
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29 Nov 2012
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"Trig" redirects here. For other uses, see Trig (disambiguation).
Trigonometry (from Greek trigōnon "triangle" + metron "measure"[1]) is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies.[2] It is also the foundation of the practical art of surveying.
Trigonometry basics are often taught in school either as a separate course or as part of a precalculus course. The trigonometric functions are pervasive in parts of pure mathematics and applied mathematics such as Fourier analysis and the wave equation, which are in turn essential to many branches of science and technology. Spherical trigonometry studies triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. It is fundamental to astronomy and navigation. Trigonometry on surfaces of negative curvature is part of Hyperbolic geometry.
Contents |
History[link]
Main article: History of trigonometry
Sumerian astronomers introduced angle measure, using a division of circles into 360 degrees.[4] They and their successors the Babylonians studied the ratios of the sides of similar triangles and discovered some properties of these ratios, but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a similar methodology.[5] The ancient Greeks transformed trigonometry into an ordered science.[6]
Classical Greek mathematicians (such as Euclid and Archimedes) studied the properties of chords and inscribed angles in circles, and proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. Claudius Ptolemy expanded upon Hipparchus' Chords in a Circle in his Almagest.[7] The modern sine function was first defined in the Surya Siddhanta, and its properties were further documented by the 5th century Indian mathematician and astronomer Aryabhata.[8] These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry.[citation needed] At about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. Knowledge of trigonometric functions and methods reached Europe via Latin translations of the works of Persian and Arabic astronomers such as Al Battani and Nasir al-Din al-Tusi.[9] One of the earliest works on trigonometry by a European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus. Trigonometry was still so little known in 16th century Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explaining its basic concepts.
Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics.[10] Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595.[11] Gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The works of James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series.[12] Also in the 18th century, Brook Taylor defined the general Taylor series.[13]
Overview[link]
If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:
- Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): \sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): \cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.
- Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): \tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{\sin A}{\cos A}\,.
The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. Many English speakers find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA (see below under Mnemonics).
The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec), and cotangent (cot), respectively:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \csc A=\frac{1}{\sin A}=\frac{c}{a} ,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \sec A=\frac{1}{\cos A}=\frac{c}{b} ,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \cot A=\frac{1}{\tan A}=\frac{\cos A}{\sin A}=\frac{b}{a} .
The inverse functions are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities. The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-".
With these functions one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known. These laws are useful in all branches of geometry, since every polygon may be described as a finite combination of triangles.
Extending the definitions[link]
File:Sin-cos-defn-I.png
Fig. 1a - Sine and cosine of an angle θ defined using the unit circle.
The above definitions apply to angles between 0 and 90 degrees (0 and π/2 radians) only. Using the unit circle, one can extend them to all positive and negative arguments (see trigonometric function). The trigonometric functions are periodic, with a period of 360 degrees or 2π radians. That means their values repeat at those intervals. The tangent and cotangent functions also have a shorter period, of 180 degrees or π radians.
The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus and infinite series. With these definitions the trigonometric functions can be defined for complex numbers. The complex exponential function is particularly useful.
- Failed to parse (Missing texvc executable; please see math/README to configure.): e^{x+iy} = e^x(\cos y + i \sin y).
See Euler's and De Moivre's formulas.
-
Graphing process of y = sin(x) using a unit circle.
-
Graphing process of y = tan(x) using a unit circle.
-
Graphing process of y = csc(x) using a unit circle.
[edit] Mnemonics
Main article: Mnemonics in trigonometry
A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters. For instance, a mnemonic for English speakers is SOH-CAH-TOA:
- Sine = Opposite ÷ Hypotenuse
- Cosine = Adjacent ÷ Hypotenuse
- Tangent = Opposite ÷ Adjacent
One way to remember the letters is to sound them out phonetically (i.e. "SOH-CAH-TOA", which is pronounced 'so-kə-tow'-uh').[14] Another method is to expand the letters into a sentence, such as "Some Old Hippy Caught Another Hippy Trippin' On Acid".[15] or "Some Old Houses, Can't Always Hide, Their Old Age"
Calculating trigonometric functions[link]
Main article: Generating trigonometric tables
Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions.
Today scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan and sometimes cis) and their inverses. Most allow a choice of angle measurement methods: degrees, radians and, sometimes, grad.[citation needed] Most computer programming languages provide function libraries that include the trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers have built-in instructions for calculating trigonometric functions.[citation needed]
Applications of trigonometry[link]
Main article: Uses of trigonometry
There are an enormous number of uses of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves.
Fields that use trigonometry or trigonometric functions include astronomy (especially for locating apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.
Standard identities[link]
Identities are those equations that hold true for any value.
- Failed to parse (Missing texvc executable; please see math/README to configure.): \sin^2 A + \cos^2 A = 1 \
- Failed to parse (Missing texvc executable; please see math/README to configure.): \sec^2 A - \tan^2 A = 1 \
- Failed to parse (Missing texvc executable; please see math/README to configure.): \csc^2 A - \cot^2 A = 1 \
Angle transformation formulas[link]
- Failed to parse (Missing texvc executable; please see math/README to configure.): \sin (A + B) = \sin A \ \cos B + \cos A \ \sin B
- Failed to parse (Missing texvc executable; please see math/README to configure.): \sin (A - B) = \sin A \ \cos B - \cos A \ \sin B
- Failed to parse (Missing texvc executable; please see math/README to configure.): \cos (A + B) = \cos A \ \cos B - \sin A \ \sin B
- Failed to parse (Missing texvc executable; please see math/README to configure.): \cos (A - B) = \cos A \ \cos B + \sin A \ \sin B
- Failed to parse (Missing texvc executable; please see math/README to configure.): \tan (A + B) = \frac{ \tan A + \tan B }{ 1 - \tan A \ \tan B}
- Failed to parse (Missing texvc executable; please see math/README to configure.): \tan (A - B) = \frac{ \tan A - \tan B }{ 1 + \tan A \ \tan B}
Common formulas[link]
File:Triangle ABC with Sides a b c.png
Triangle with sides a,b,c and respectively opposite angles A,B,C
Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Some identities equate an expression to a different expression involving the same angles. These are listed in List of trigonometric identities. Triangle identities that relate the sides and angles of a given triangle are listed below.
In the following identities, A, B and C are the angles of a triangle and a, b and c are the lengths of sides of the triangle opposite the respective angles.
Law of sines[link]
The law of sines (also known as the "sine rule") for an arbitrary triangle states:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,
where R is the radius of the circumscribed circle of the triangle:
- Failed to parse (Missing texvc executable; please see math/README to configure.): R = \frac{abc}{\sqrt{(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}.
Another law involving sines can be used to calculate the area of a triangle. Given two sides and the angle between the sides, the area of the triangle is:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mbox{Area} = \frac{1}{2}a b\sin C.
Law of cosines[link]
The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:
- Failed to parse (Missing texvc executable; please see math/README to configure.): c^2=a^2+b^2-2ab\cos C ,\,
or equivalently:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \cos C=\frac{a^2+b^2-c^2}{2ab}.\,
Law of tangents[link]
The law of tangents:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{a-b}{a+b}=\frac{\tan\left[\tfrac{1}{2}(A-B)\right]}{\tan\left[\tfrac{1}{2}(A+B)\right]}
Euler's formula[link]
Euler's formula, which states that Failed to parse (Missing texvc executable; please see math/README to configure.): e^{ix} = \cos x + i \sin x , produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \sin x = \frac{e^{ix} - e^{-ix}}{2i}, \qquad \cos x = \frac{e^{ix} + e^{-ix}}{2}, \qquad \tan x = \frac{i(e^{-ix} - e^{ix})}{e^{ix} + e^{-ix}}.
See also[link]
References[link]
- ^ "trigonometry". Online Etymology Dictionary. http://www.etymonline.com/index.php?search=trigonometry.
- ^ R. Nagel (ed.), Encyclopedia of Science, 2nd Ed., The Gale Group (2002)
- ^ Boyer (1991). "Greek Trigonometry and Mensuration". p. 162.
- ^ Aaboe, Asger. Episodes from the Early History of Astronomy. New York: Springer, 2001. ISBN 0-387-95136-9
- ^ A history of ancient mathematical astronomy: in three parts. By Otto Neugebauer. pg. 744
- ^ "The Beginnings of Trigonometry". Rutgers, The State University of New Jersey.
- ^ Marlow Anderson, Victor J. Katz, Robin J. Wilson (2004)."Sherlock Holmes in Babylon: and other tales of mathematical history". MAA. p.36. ISBN 0-88385-546-1
- ^ Boyer p215
- ^ Boyer p237, p274
- ^ Grattan-Guinness, Ivor (1997). The Rainbow of Mathematics: A History of the Mathematical Sciences. W.W. Norton. ISBN 0-393-32030-8.
- ^ Groundbreaking Scientific Experiments, Inventions, and Discoveries
- ^ William Bragg Ewald (2008)."From Kant to Hilbert: a source book in the foundations of mathematics". Oxford University Press US. p.93. ISBN 0-19-850535-3
- ^ Kelly Dempski (2002)."Focus on Curves and Surfaces". p.29. ISBN 1-59200-007-X
- ^ Weisstein, Eric W., "SOHCAHTOA" from MathWorld.
- ^ Sentences more appropriate for high schools are, "Some old horse came a'hopping through our alley" Foster, Jonathan K. (2008). Memory: A Very Short Introduction. Oxford. p. 128. ISBN 0-19-280675-0.
Bibliography[link]
- Boyer, Carl B. (1991). A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc.. ISBN 0-471-54397-7.
- Christopher M. Linton (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy . Cambridge University Press.
- Weisstein, Eric W. "Trigonometric Addition Formulas". Wolfram MathWorld. Weiner.
External links[link]
| Find more about Trigonometry on Wikipedia's sister projects: | |
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Textbooks from Wikibooks |
- Trigonometric Delights, by Eli Maor, Princeton University Press, 1998. Ebook version, in PDF format, full text presented.
- Trigonometry by Alfred Monroe Kenyon and Louis Ingold, The Macmillan Company, 1914. In images, full text presented.
- Benjamin Banneker's Trigonometry Puzzle at Convergence
- Dave's Short Course in Trigonometry by David Joyce of Clark University
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This article is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License, which means that you can copy and modify it as long as the entire work (including additions) remains under this license.
