Wikibooks has a book on the topic of

Trigonometry

The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. Calculating the final position of the astronaut at the end of the arm requires repeated use of the trigonometric functions of those angles.

All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.

Trigonometry (from Greek trigōnon "triangle" + metron "measure") ^[1] is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has 90 degrees (right triangles). Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships. Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ... Image File history File links Download high resolution version (3032x2005, 2659 KB) original description: Astronaut Stephen K. Robinson, STS-114 mission specialist, anchored to a foot restraint on the International Space Station’s Canadarm2, participates in the mission’s third session of extravehicular activity (EVA). ... Image File history File links Download high resolution version (3032x2005, 2659 KB) original description: Astronaut Stephen K. Robinson, STS-114 mission specialist, anchored to a foot restraint on the International Space Station’s Canadarm2, participates in the mission’s third session of extravehicular activity (EVA). ... ISS Canadarm2 (NASA) The Mobile Servicing System (MSS) is a robotic arm and associated equipment on the International Space Station that plays a key role in station assembly and maintenance: moving equipment and supplies around the station, supporting astronauts working in space, and servicing instruments and other payloads attached to... ISS redirects here. ... Image File history File links Circle-trig6. ... Image File history File links Circle-trig6. ... Sine redirects here. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... This article is about the mathematical construct. ... This article is about angles in geometry. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

Trigonometry has applications in both pure mathematics and in applied mathematics, where it is essential in many branches of science and technology. It is usually taught in secondary schools either as a separate course or as part of a precalculus course. Trigonometry is informally called “trig.” Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. ... Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ... High School also refers to the highest form of classical riding, High School Dressage. ... In mathematics education, precalculus, an advanced form of secondary school algebra, is a foundational mathematical discipline. ...

A branch of trigonometry, called spherical trigonometry, studies triangles on spheres, and is important in astronomy and navigation. Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... For other uses, see Sphere (disambiguation). ... For other uses, see Astronomy (disambiguation). ... This article is about determination of position and direction on or above the surface of the earth. ...

History

Table of Trigonometry, 1728 Cyclopaedia

Main article: History of trigonometric functions

Trigonometry was probably developed for use in sailing as a navigation method used with astronomy.^[2] The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago.^{[citation needed]} The common practice of measuring angles in degrees, minutes and seconds comes from the Babylonian's base sixty system of numeration. The Sulba Sutras written in India, between 800 BC and 500 BC, correctly computes the sine of π/4 (45°) as 1/√2 in a procedure for circling the square (the opposite of squaring the circle). Image File history File linksMetadata Download high resolution version (850x2607, 683 KB) This article incorporates content from the 1728 Cyclopaedia, a publication in the public domain. ... Image File history File linksMetadata Download high resolution version (850x2607, 683 KB) This article incorporates content from the 1728 Cyclopaedia, a publication in the public domain. ... 1913 advertisement for Encyclopædia Britannica. ... The history of trigonometric functions may span about 4000 years. ... Khafres Pyramid and the Great Sphinx of Giza, built about 2550 BC during the Fourth Dynasty of the Old Kingdom,[1] are enduring symbols of the civilization of ancient Egypt Ancient Egypt was a civilization in Northeastern Africa concentrated along the middle to lower reaches of the Nile River... Mesopotamia was a cradle of civilization geographically located between the Tigris and Euphrates rivers, largely corresponding to modern-day Iraq. ... Excavated ruins of Mohenjo-daro. ... Babylonia was a state in southern Mesopotamia, in modern Iraq, combining the territories of Sumer and Akkad. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ... Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ... This article describes the unit of angle. ... Squaring the circle: the areas of this square and this circle are equal. ...

The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus[1] circa 150 BC, who compiled a trigonometric table using the sine for solving triangles. Ptolemy further developed trigonometric calculations circa 100 AD. Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. ... For the Athenian tyrant, see Hipparchus (son of Pisistratus). ... Before calculators were cheap and plentiful, people would use mathematical tables —lists of numbers showing the results of calculation with varying variables— to simplify and drastically speed up computation. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... This article is about the geographer, mathematician and astronomer Ptolemy. ...

The ancient Sinhalese in Sri Lanka, when constructing reservoirs in the Anuradhapura kingdom, used trigonometry to calculate the gradient of the water flow. Archeological research also provides evidence of trigonometry used in other unique hydrological structures dating back to 4 BC.^[3] Language(s) Sinhala Religion(s) Theravada Buddhism, Christianity, small groups of atheists, agnostics, Muslims, others Related ethnic groups Indo-Aryans, Dravidians, Veddahs, Bengalis The Sinhalese are the main ethnic group of Sri Lanka. ... Anuradhapura, ( in Sinhala), is one of the ancient capitals of Sri Lanka, world famous for its well preserved ruins of the Great Sri Lankan Civilization. ...

The Indian mathematician Aryabhata in 499, gave tables of half chords which are now known as sine tables, along with cosine tables. He used zya for sine, kotizya for cosine, and otkram zya for inverse sine, and also introduced the versine. Another Indian mathematician, Brahmagupta in 628, used an interpolation formula to compute values of sines, up to the second order of the Newton-Stirling interpolation formula. For other uses, see Aryabhata (disambiguation). ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... The versed sine, also called the versine and, in Latin, the sinus versus (flipped sine) or the sagitta (arrow), is a trigonometric function versin(θ) (sometimes further abbreviated vers) defined by the equation: versin(θ) = 1 − cos(θ) = 2 sin2(θ / 2) There are also three corresponding functions: the coversed... Brahmagupta (ब्रह्मगुप्त) ( ) (589–668) was an Indian mathematician and astronomer. ... In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... James Stirling (April 22, 1692–December 5, 1770) was a Scottish mathematician. ...

In the 10th century, the Persian mathematician and astronomer Abul Wáfa introduced the tangent function and improved methods of calculating trigonometry tables. He established the angle addition identities, e.g. sin (a + b), and discovered the sine formula for spherical geometry: Abul Wafa Muhammad Ibn Muhammad Ibn Yahya Ibn Ismail Buzjani (940 – 997 or 998) was a Persian mathematician and astronomer. ...

Also in the late 10th and early 11th centuries, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the formula Ibn Yunus (Arabic: ابن يونس) (full name, Abu al-Hasan Ali abi Said Abd al-Rahman ibn Ahmad ibn Yunus al-Sadafi al-Misri) (c. ...

.

Indian mathematicians were the pioneers of variable computations algebra for use in astronomical calculations along with trigonometry. Lagadha (circa 1350-1200 BC) is the first person thought to have used geometry and trigonometry for astronomy, in his Vedanga Jyotisha. This article is under construction. ... This article is about the branch of mathematics. ... Lagadha (लगध) is the author of Vedanga Jyotisha, the text on Vedic astronomy that has been dated to 1350 BC. This text describes rules for tracking the motions of the sun and the moon. ... The Vedanga Jyotisha, is an Indian text on Jyotisha (Hindu astronomy), redacted by Lagadha (लगध). The text is foundational to the Jyotisha discipline of Vedanga, and is dated to the final centuries BCE.[1] The text describes rules for tracking the motions of the sun and the moon. ...

Persian mathematician Omar Khayyám (1048-1131) combined trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means. Khayyam solved the cubic equation x³ + 200x = 20x² + 2000 and found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables. This article is about the Persian people, an ethnic group found mainly in Iran. ... In the history of mathematics, Islamic mathematics or Arabic mathematics refers to the mathematics developed by the Islamic civilization between 622 and 1600. ... For other people, places or with similar names of Khayam, see Khayyam (disambiguation). ... In mathematics, approximation theory is concerned with how functions can be approximated with other, simpler, functions, and with characterising in a quantitative way the errors introduced thereby. ... In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ...

Detailed methods for constructing a table of sines for any angle were given by the Indian mathematician Bhaskara in 1150, along with some sine and cosine formulae. Bhaskara also developed spherical trigonometry. Bhaskara (1114-1185), also known as Bhaskara II and Bhaskara Achārya (Bhaskara the teacher), was an Indian mathematician-astronomer. ... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ...

The 13th century Persian mathematician Nasir al-Din Tusi, along with Bhaskara, was probably the first to treat trigonometry as a distinct mathematical discipline. Nasir al-Din Tusi in his Treatise on the Quadrilateral was the first to list the six distinct cases of a right angled triangle in spherical trigonometry. This article is about the Persian people, an ethnic group found mainly in Iran. ... In the history of mathematics, Islamic mathematics or Arabic mathematics refers to the mathematics developed by the Islamic civilization between 622 and 1600. ... Nasir Tusi Abu Jafar Muhammad Ibn Muhammad Ibn al-Hasan Nasir al-Din al-Tusi (1201–1274) was a Persian scientist, of Shia Islamic belief, born in Tus, Khorasan, Iran. ...

In the 14th century, Persian mathematician al-Kashi and Timurid mathematician Ulugh Beg (grandson of Timur) produced tables of trigonometric functions as part of their studies of astronomy. Kashani, dubbed, the Second Ptolemy, was an outstanding Persian mathematician of the middle ages. ... Timurid can refer to several entities, related to Timur: Timurid Dynasty Timurid Empire Timurid Emirates This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Ulugh Beg, here depicted on a Soviet stamp, was one of Islams greatest astronomers during the Middle Ages. ... For the similar-sounding word Timor, see Timor (disambiguation). ...

The mathematician Bartholemaeus Pitiscus published an influential work on trigonometry in 1595 which may have coined the word "trigonometry". For the crater, see Pitiscus (crater). ...

Overview

In this right triangle: sin A = a/c; cos A = b/c; tan A = a/b.

If one angle of a right 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 right triangle is completely determined, up to similarity, by the angles. This means that once one of the other angles is known, the ratios of the various sides are always the same regardless of the overall size of the triangle. These ratios are given by the following trigonometric functions of the known angle A: Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... This article is about angles in geometry. ... A pair of complementary angles, because they add up to 90 degrees. ... Look up shape in Wiktionary, the free dictionary. ... // Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. ... This article is about the mathematical concept. ... Sine redirects here. ...

• The sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.

• The cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.

• The tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.

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 people 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. 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 reciprocal function: y = 1/x. ... In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. ... In mathematics, trigonometric identities (or trig identities for short) are equations involving trigonometric functions that are true for all values of the occurring variables. ...

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 an 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. In trigonometry, the law of sines (or sine law, sine formula) is a statement about arbitrary triangles in the plane. ... Fig. ... Look up polygon in Wiktionary, the free dictionary. ...

Extending the definitions

Graphs of the functions sin(x) and cos(x), where the angle x is measured in radians.

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.

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. Image File history File links Sine_cosine_plot. ... Image File history File links Sine_cosine_plot. ... Some common angles, measured in radians. ... Illustration of a unit circle. ... Sine redirects here. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ...

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 function cis is particularly useful For other uses, see Calculus (disambiguation). ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...

See Euler's and De Moivre's formulas. Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... de Moivres formula, named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and any integer n it holds that The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. ...

Mnemonics

Students often use mnemonics 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, as in SOH-CAH-TOA.
For other uses, see Mnemonic (disambiguation). ...

Sine = Opposite ÷ Hypotenuse

Cosine = Adjacent ÷ Hypotenuse

Tangent = Opposite ÷ Adjacent

Alternatively, one can devise sentences which consist of words beginning with the letters to be remembered. For example, to recall that Tan = Opposite/Adjacent, the letters T-O-A must be remembered. Any memorable phrase constructed of words beginning with the letters T-O-A will serve.

Another type of mnemonic describes facts in a simple, memorable way, such as "Plus to the right, minus to the left; positive height, negative depth," which refers to trigonometric functions generated by a revolving line.

Calculating trigonometric functions

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. Tables of trigonometric functions are useful in a number of areas. ... Before calculators were cheap and plentiful, people would use mathematical tables —lists of numbers showing the results of calculation with varying variables— to simplify and drastically speed up computation. ... This article is about interpolation in mathematics. ... A typical 10 inch student slide rule (Pickett N902-T simplex trig). ...

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. 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. A basic arithmetic calculator. ... The grad is a measurement of plane angles of value 1/400 of a full circle, thus dividing a right angle in 100. ... A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ... A floating point unit (FPU) is a part of a computer system specially designed to carry out operations on floating point numbers. ...

Applications of trigonometry

Main article: Uses of trigonometry

There are an enormous number of applications 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. Trigonometry has an enormous variety of applications. ... Trigonometry has an enormous variety of applications. ... Triangulation can be used to find the distance from the shore to the ship. ... For other uses, see Astronomy (disambiguation). ... It has been suggested that this article or section be merged into Global Navigation Satellite System. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... For other uses, see Light (disambiguation). ...

Fields which make use of trigonometry or trigonometric functions include astronomy (especially, for locating the 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. For other uses, see Astronomy (disambiguation). ... This article is about determination of position and direction on or above the surface of the earth. ... Music theory is a field of study that investigates the nature or mechanics of music. ... Acoustics is the branch of physics concerned with the study of sound (mechanical waves in gases, liquids, and solids). ... For the book by Sir Isaac Newton, see Opticks. ... This article is about the engineering discipline. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... This article is about the field of statistics. ... For the song by Girls Aloud see Biology (song) Biology studies the variety of life (clockwise from top-left) E. coli, tree fern, gazelle, Goliath beetle Biology (from Greek: βίος, bio, life; and λόγος, logos, speech lit. ... Medical imaging designates the ensemble of techniques and processes used to create images of the human body (or parts thereof) for clinical purposes (medical procedures seeking to reveal, diagnose or examine disease) or medical science (including the study of normal anatomy and function). ... CAT apparatus in a hospital Computed axial tomography (CAT), computer-assisted tomography, computed tomography, CT, or body section roentgenography is the process of using digital processing to generate a three-dimensional image of the internals of an object from a large series of two-dimensional X-ray images taken around... For other uses, see Ultrasound (disambiguation). ... For other uses, see Pharmacy (disambiguation). ... For other uses, see Chemistry (disambiguation). ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Cryptology is an umbrella term for cryptography and cryptanalysis. ... Seismology (from the Greek seismos = earthquake and logos = word) is the scientific study of earthquakes and the propagation of elastic waves through the Earth. ... // Meteorology (from Greek: μετέωρον, meteoron, high in the sky; and λόγος, logos, knowledge) is the interdisciplinary scientific study of the atmosphere that focuses on weather processes and forecasting. ... Thermohaline circulation Oceanography (from Ocean + Greek γράφειν = write), also called oceanology or marine science, is the branch of Earth Sciences that studies the Earths oceans and seas. ... == Headline text ==cant there be some kind of picture somewhere so i can see by picture???? Physical science is a encompassing term for the branches of natural science, and science, that study non-living systems, in contrast to the biological sciences. ... Surveyor at work with a leveling instrument. ... An old geodetic pillar (1855) at Ostend, Belgium A Munich archive with lithography plates of maps of Bavaria Geodesy (pronounced [1]), also called geodetics, a branch of earth sciences, is the scientific discipline that deals with the measurement and representation of the Earth, including its gravity field, in a three... This article is about building architecture. ... Phonetics (from the Greek word φωνή, phone meaning sound or voice) is the study of the sounds of human speech. ... Face-to-face trading interactions on the New York Stock Exchange trading floor. ... Electrical Engineers design power systems… … and complex electronic circuits. ... Mechanical Engineering is an engineering discipline that involves the application of principles of physics for analysis, design, manufacturing, and maintenance of mechanical systems. ... The Falkirk Wheel in Scotland. ... This article is about the scientific discipline of computer graphics. ... Cartography or mapmaking (in Greek chartis = map and graphein = write) is the study and practice of making maps or globes. ... Crystallography (from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein = write) is the experimental science of determining the arrangement of atoms in solids. ... Game development - Wikipedia, the free encyclopedia /**/ @import /skins-1. ...

Marine sextants like this are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can then be determined from several such measurements.

Image File history File linksMetadata Download high-resolution version (3072x2304, 3716 KB) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Sextant Trigonometry Metadata This file contains additional information, probably added from the digital camera or scanner used to... Image File history File linksMetadata Download high-resolution version (3072x2304, 3716 KB) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Sextant Trigonometry Metadata This file contains additional information, probably added from the digital camera or scanner used to... A sextant is a measuring instrument generally used to measure the angle of elevation of a celestial object above the horizon. ... A marine chronometer is a timekeeper precise enough to be used as a portable time standard, used to determine longitude by means of celestial navigation. ...

Common formulae

Main article: Trigonometric identity

Main article: Trigonometric function

Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Many express important geometric relationships. For example, the Pythagorean identities are an expression of the Pythagorean Theorem. Here are some of the more commonly used identities, as well as the most important formulae connecting angles and sides of an arbitrary triangle. For more identities see trigonometric identity. In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... Sine redirects here. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

Trigonometric identities

Pythagorean identities

Sum and product identities

Sum to product:

Product to sum:

Sine, cosine, and tangent of a sum

Detailed, diagramed proofs of the first two of these formulas are available
for download as a four-page PDF document at Image:Sine Cos Proofs.pdf.

Double-angle identities

Half-angle identities

Note that is correct, it means it may be either one, depending on the value of A/2.

Stereographic ( or parametric ) identities

where .

Triangle identities

Laws of Sines and Cosines

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. Image File history File links Size of this preview: 712 × 599 pixelsFull resolution (898 × 756 pixels, file size: 12 KB, MIME type: image/png) Any triangle A B C with sides a, b and c each side opposite to the angle with the same letter in capital form I, the... Image File history File links Size of this preview: 712 × 599 pixelsFull resolution (898 × 756 pixels, file size: 12 KB, MIME type: image/png) Any triangle A B C with sides a, b and c each side opposite to the angle with the same letter in capital form I, the...

Law of sines

The law of sines (also know as the "sine rule") for an arbitrary triangle states: In trigonometry, the law of sines (or sine law, sine formula) is a statement about arbitrary triangles in the plane. ...

where R is the radius of the circumcircle of the triangle. In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ...

Law of cosines

The law of cosines (also known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles: Fig. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...

or equivalently:

Law of tangents

The law of tangents: In trigonometry, the law of tangents is a statement about arbitrary triangles in the plane. ...

See also

• Uses of trigonometry
• Trigonometric functions
• List of basic trigonometry topics
• Trigonometric identity
• Trigonometry in Galois fields
• List of triangle topics

Trigonometry has an enormous variety of applications. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... Trigonometry is a branch of mathematics which deals with angles, triangles and trigonometric functions such as sine, cosine and tangent. ... In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

References

External links

Find more about Trigonometry on Wikipedia's sister projects:
	Dictionary definitions
	Textbooks
	Quotations
	Source texts
	Images and media
	News stories
	Learning resources

• 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.
• Trigonometry on PlainMath.net Trigonometry Articles from PlainMath.Net
• Trigonometry on Mathwords.com index of trigonometry entries on Mathwords.com
• Benjamin Banneker's Trigonometry Puzzle at Convergence
• Trigonometry
• Dave's Short Course in Trigonometry by David Joyce of Clark University