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.
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.
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.
TriangleA triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted . In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane.
Sine and cosineIn mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle , the sine and cosine functions are denoted simply as and .
Right triangleA right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle (ὀρθόσγωνία), is a triangle in which one angle is a right angle (that is, a 90-degree angle), i.e., in which two sides are perpendicular. The relation between the sides and other angles of the right triangle is the basis for trigonometry. The side opposite to the right angle is called the hypotenuse (side c in the figure).
Exact trigonometric valuesIn mathematics, the values of the trigonometric functions can be expressed approximately, as in , or exactly, as in . While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots. The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 90°.
Special right triangleA special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio.
Integer triangleAn integer triangle or integral triangle is a triangle all of whose side lengths are integers. A rational triangle is one whose side lengths are rational numbers; any rational triangle can be rescaled by the lowest common denominator of the sides to obtain a similar integer triangle, so there is a close relationship between integer triangles and rational triangles. Sometimes other definitions of the term rational triangle are used: Carmichael (1914) and Dickson (1920) use the term to mean a Heronian triangle (a triangle with integral or rational side lengths and area);cite book |last=Carmichael |first=R.
LausanneLausanne (loʊˈzæn , USalsoloʊˈzɑːn , lɔzan; Losena lɔˈzəna) is the capital and largest city of the Swiss French-speaking canton of Vaud. It is a hilly city situated on the shores of Lake Geneva, about halfway between the Jura Mountains and the Alps, and facing the French town of Évian-les-Bains across the lake. Lausanne is located northeast of Geneva, the nearest major city. The municipality of Lausanne has a population of about 140,000, making it the fourth largest city in Switzerland after Basel, Geneva, and Zurich, with the entire agglomeration area having about 420,000 inhabitants (as of January 2019).
Equilateral triangleIn geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.
Lausanne DistrictLausanne District (District de Lausanne) is a district in the canton of Vaud, Switzerland. The seat of the district is the city of Lausanne. Lausanne has an area, , of . Of this area, or 27.3% is used for agricultural purposes, while or 30.9% is forested. Of the rest of the land, or 41.6% is settled (buildings or roads) and or 0.1% is unproductive land. Lausanne has a population () of . Most of the population () speaks French (159,559 or 79.6%), with German being second most common (8,534 or 4.
Romanel-sur-LausanneRomanel-sur-Lausanne (ʁɔmanɛl syʁ lɔzan, literally Romanel on Lausanne; Romanél) is a municipality in the canton of Vaud in Switzerland, located in the district of Lausanne. Romanel-sur-Lausanne is first mentioned in 1184 as Romanes. Romanel-sur-Lausanne has an area, , of (depending on calculation method). Of this area, or 59.7% is used for agricultural purposes, while or 3.8% is forested. Of the rest of the land, or 37.8% is settled (buildings or roads). Of the built up area, industrial buildings made up 5.
Cheseaux-sur-LausanneCheseaux-sur-Lausanne (ʃəzo syʁ lɔzan, literally Cheseaux on Lausanne; Chesâls) is a municipality in the district of Lausanne in the canton of Vaud in Switzerland. It is a suburb of the city of Lausanne. Cheseaux-sur-Lausanne is first mentioned in 1228 as Chesaus. Cheseaux-sur-Lausanne has an area, , of . Of this area, or 60.3% is used for agricultural purposes, while or 14.6% is forested. Of the rest of the land, or 24.0% is settled (buildings or roads), or 0.7% is either rivers or lakes.
Exponential functionThe exponential function is a mathematical function denoted by or (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers.
Belmont-sur-LausanneBelmont-sur-Lausanne (bɛlmɔ̃ syʁ lɔzan, literally Belmont on Lausanne) is a municipality in the district of Lavaux-Oron in the canton of Vaud in Switzerland. It is a suburb of the city of Lausanne. Belmont-sur-Lausanne is first mentioned in 1228 as Belmunt sowie apud bellum Montem. Belmont-sur-Lausanne has an area, , of . Of this area, or 35.1% is used for agricultural purposes, while or 29.8% is forested. Of the rest of the land, or 33.2% is settled (buildings or roads). Of the built up area, housing and buildings made up 23.
Logarithmic scaleA logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way. As opposed to a linear number line in which every unit of distance corresponds to adding by the same amount, on a logarithmic scale, every unit of length corresponds to multiplying the previous value by the same amount. Hence, such a scale is nonlinear: the numbers 1, 2, 3, 4, 5, and so on, are not equally spaced. Rather, the numbers 10, 100, 1000, 10000, and 100000 would be equally spaced.
Triangle centerIn geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions. Each of these classical centers has the property that it is invariant (more precisely equivariant) under similarity transformations.
Altitude (triangle)In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the the side opposite the vertex. This line containing the opposite side is called the extended base of the altitude. The intersection of the extended base and the altitude is called the foot of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as dropping the altitude at that vertex.
