Trigonometric functionsIn mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
Inverse trigonometric functionsIn mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
TrigonometryTrigonometry () is a branch of mathematics concerned with relationships between angles and ratios of lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.
List of trigonometric identitiesIn trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle. These identities are useful whenever expressions involving trigonometric functions need to be simplified.
Hyperbolic functionsIn mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry.
Algebraic expressionIn mathematics, an algebraic expression is an expression built up from constant algebraic numbers, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For example, 3x2 − 2xy + c is an algebraic expression. Since taking the square root is the same as raising to the power 1/2, the following is also an algebraic expression: An algebraic equation is an equation involving only algebraic expressions.
Algebraic geometryAlgebraic geometry is a branch of mathematics which classically studies zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations.
AlgebraAlgebra () is the study of variables and the rules for manipulating these variables in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields.
Space (mathematics)In mathematics, a space is a set (sometimes called a universe) with some added structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself. A space consists of selected mathematical objects that are treated as points, and selected relationships between these points. The nature of the points can vary widely: for example, the points can be elements of a set, functions on another space, or subspaces of another space.
Algebraic number fieldIn mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension has finite degree (and hence is an algebraic field extension). Thus is a field that contains and has finite dimension when considered as a vector space over . The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
X-rayX-ray radiation, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 nanometers to 10 picometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz (3e16Hz to 3e19Hz) and energies in the range 124 keV to 145 eV, respectively. X-ray wavelengths are shorter than those of UV rays and typically longer than those of gamma rays.
Style (form of address)A style of office or form of address, also called manner of address, is an official or legally recognized form of address for a person or other entity (such as a government or company), and may often be used in conjunction with a personal title. A style, by tradition or law, precedes a reference to a person who holds a post or political office, and is sometimes used to refer to the office itself. An honorific can also be awarded to an individual in a personal capacity.
X-ray tubeAn X-ray tube is a vacuum tube that converts electrical input power into X-rays. The availability of this controllable source of X-rays created the field of radiography, the imaging of partly opaque objects with penetrating radiation. In contrast to other sources of ionizing radiation, X-rays are only produced as long as the X-ray tube is energized. X-ray tubes are also used in CT scanners, airport luggage scanners, X-ray crystallography, material and structure analysis, and for industrial inspection.
PrinceA prince is a male ruler (ranked below a king, grand prince, and grand duke) or a male member of a monarch's or former monarch's family. Prince is also a title of nobility (often highest), often hereditary, in some European states. The female equivalent is a princess. The English word derives, via the French word prince, from the Latin noun prīnceps, from primus (first) and caput (head), meaning "the first, foremost, the chief, most distinguished, noble ruler, prince".
SinIn a religious context, 'sin' is a transgression against divine law or a law of God. Each culture has its own interpretation of what it means to commit a sin. While sins are generally considered actions, any thought, word, or act considered immoral, selfish, shameful, harmful, or alienating might be termed "sinful".
Mortal sinA mortal sin (peccatum mortale), in Catholic theology, is a gravely sinful act which can lead to damnation if a person does not repent of the sin before death. It is alternatively called deadly, grave, and serious. A sin is considered to be "mortal" when its quality is such that it leads to a separation of that person from God's saving grace. Three conditions must together be met for a sin to be mortal: "Mortal sin is sin whose object is grave matter and which is also committed with full knowledge and deliberate consent.
Christian views on sinIn Christianity, 'sin' is an immoral act considered to be a transgression of divine law. The doctrine of sin is central to the Christian faith, since its basic message is about redemption in Christ. Hamartiology, a branch of Christian theology which is the study of sin, describes sin as an act of offence against God by despising his persons and Christian biblical law, and by injuring others. Christian hamartiology is closely related to concepts of natural law, moral theology and Christian ethics.
X-ray astronomyX-ray astronomy is an observational branch of astronomy which deals with the study of X-ray observation and detection from astronomical objects. X-radiation is absorbed by the Earth's atmosphere, so instruments to detect X-rays must be taken to high altitude by balloons, sounding rockets, and satellites. X-ray astronomy uses a type of space telescope that can see x-ray radiation which standard optical telescopes, such as the Mauna Kea Observatories, cannot.
Eternal sinIn Christian hamartiology, eternal sin, the unforgivable sin, unpardonable sin, or ultimate sin is the sin which will not be forgiven by God. One eternal or unforgivable sin (blasphemy against the Holy Spirit), also known as the sin unto death, is specified in several passages of the Synoptic Gospels, including Mark 3:28–29, Matthew 12:31–32, and Luke 12:10, as well as other New Testament passages including Hebrews 6:4–6, Hebrews 10:26–31, and 1 John 5:16.
FurFur is a thick growth of hair that covers the skin of almost all mammals. It consists of a combination of oily guard hair on top and thick underfur beneath. The guard hair keeps moisture from reaching the skin; the underfur acts as an insulating blanket that keeps the animal warm. The fur of mammals has many uses: protection, sensory purposes, waterproofing, and camouflaging, with the primary usage being thermoregulation.
