Geometric foundations of trigonometry

• Altern base
  Altern base
  In trigonometry, the altern base is distinguished from the true base. Thus, in oblique triangles, the true base is the sum of the sides, and then the difference of the sides is the altern base; or the true base is the difference of the sides, and then the sum of the side is the altern base....
  
  
• Angle
  Angle
  In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
  
  
• Angular distance
  Angular distance
  In mathematics and all natural sciences , the angular distance between two point objects, as observed from a location different from either of these objects, is the size of the angle between the two directions originating from...
  
  
• Angle excess
  Angle excess
  Angle excess, also known as spherical excess is the amount by which the sum of the angles of a polygon on a sphere exceeds the sum of the angles of a polygon with the same number of sides in a plane. For instance, a plane triangle has an angle sum of 180°; an octant is a spherical triangle with...
  
  
• Chord (geometry)
  Chord (geometry)
  A chord of a circle is a geometric line segment whose endpoints both lie on the circumference of the circle.A secant or a secant line is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse...
  
  
• Circle
  Circle
  A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
  
   (also see List of circle topics)
  • Unit circle
    Unit circle
    In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
    
    
• Degree (angle)
  Degree (angle)
  A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
  
  
• Grad (angle)
  Grad (angle)
  The gradian is a unit of plane angle, equivalent to of a turn. It is also known as gon, grad, or grade . One grad equals of a degree or of a radian...
  
  
• Hypotenuse
  Hypotenuse
  In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the...
  
  
• Opposites theorem
• Pi
  ' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
  
   (pi)
• Ptolemy's theorem
  Ptolemy's theorem
  In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral . The theorem is named after the Greek astronomer and mathematician Ptolemy...
  
  
• Pythagorean theorem
  Pythagorean theorem
  In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
  
  
• Radian
  Radian
  Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
  
  
• Right angle
  Right angle
  In geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles...
  
  
• Thales' theorem
  Thales' theorem
  In geometry, Thales' theorem states that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Thales' theorem is a special case of the inscribed angle theorem...
  
  
• Trigonometric function
  Trigonometric function
  In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
  
  
• Triangle
  Triangle
  A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
  
   (also see List of triangle topics)

Trigonometric functions

• Sine, Cosine, Tangent, Cotangent, Secant, Cosecant – see Trigonometric function
  Trigonometric function
  In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
  
  

• atan2
  Atan2
  In trigonometry, the two-argument function atan2 is a variation of the arctangent function. For any real arguments and not both equal to zero, is the angle in radians between the positive -axis of a plane and the point given by the coordinates on it...
  
  
• cis—see Euler's formula
  Euler's formula
  Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
  
  
• Cofunction
  Cofunction
  In mathematics, a function f is cofunction of a function g if f = g whenever A and B are complementary angles...
  
  
• Exsecant
• Gudermannian function
• Inverse trigonometric functions
• Jyā, koti-jyā and utkrama-jyā
  Jyā, koti-jyā and utkrama-jyā
  Jyā, koti-jyā and utkrama-jyā are three trigonometric functions introduced by Indian astronomers and mathematicians. The earliest known Indian treatise containing references to these functions is Surya Siddhanta. These are functions of arcs of circles and not functions of angles. Jyā and...
  
  
• Versine
  Versine
  The versine or versed sine, versin, is a trigonometric function equal to and 2sin2. It appeared in some of the earliest trigonometric tables and was once widespread, but it is now little-used...
  
  

Trigonometric identities

• De Moivre's formula
  De Moivre's formula
  In mathematics, de Moivre's formula , named after Abraham de Moivre, states that for any complex number x and integer n it holds that...
  
  
• Euler's formula
  Euler's formula
  Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
  
  
• Hermite's cotangent identity
  Hermite's cotangent identity
  In mathematics, Hermite's cotangent identity is a trigonometric identity discovered by Charles Hermite. Suppose a1, ..., an are complex numbers, no two of which differ by an integer multiple of π. Let...
  
  
• Lagrange's trigonometric identities
• Morrie's law
• Proofs of trigonometric identities
  Proofs of trigonometric identities
  Proofs of trigonometric identities are used to show relations between trigonometric functions. This article will list trigonometric identities and prove them.-Definitions:Referring to the diagram at the right, the six trigonometric functions of θ are:...
  
  
• Pythagorean trigonometric identity
  Pythagorean trigonometric identity
  The Pythagorean trigonometric identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one the basic relations between the sine and cosine functions, from which all others may be derived.-Statement of...
  
  
• Tangent half-angle formula

Solution of triangles

• Law of sines
  Law of sines
  In trigonometry, the law of sines is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles...
  
  
• Law of cosines
  Law of cosines
  In trigonometry, the law of cosines relates the lengths of the sides of a plane triangle to the cosine of one of its angles. Using notation as in Fig...
  
  
• Law of tangents
  Law of tangents
  In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposite sides....
  
  
• Law of cotangents
  Law of cotangents
  In trigonometry, the law of cotangents relates the radius of the inscribed circle of a triangle to its sides and angles.When a, b, and c are the lengths of the three sides of the triangle, and α, β, and γ are the angles opposite those three respective sides...
  
  
• Mollweide's formula
  Mollweide's formula
  In trigonometry, Mollweide's formula, sometimes referred to in older texts as Mollweide's equations, named after Karl Mollweide, is a set of two relationships between sides and angles in a triangle...
  
  

More advanced trigonometric concepts and methods

• Chebyshev polynomials
  Chebyshev polynomials
  In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...
  
  
• Conway triangle notation
  Conway triangle notation
  In geometry, the Conway triangle notation, named after John Horton Conway, allows trigonometric functions of a triangle to be managed algebraically...
  
  
• Exact trigonometric constants
  Exact trigonometric constants
  Exact constant expressions for trigonometric values are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification....
  
  
• Generalized trigonometry
  Generalized trigonometry
  Ordinary trigonometry studies triangles in the euclidean plane R2. There are a number of ways of defining the ordinary euclidean geometric trigonometric functions on real numbers: right-angled triangle definitions, unit-circle definitions, series definitions, definitions via differential equations,...
  
  
• Gudermannian function
• Lissajous curve
  Lissajous curve
  In mathematics, a Lissajous curve , also known as Lissajous figure or Bowditch curve, is the graph of a system of parametric equationswhich describe complex harmonic motion...
  
  
• Polar sine
  Polar sine
  In mathematics, the polar sine of a vertex angle of a polytope is defined as follows. Let v1, ..., vn, n ≥ 2, be non-zero vectors from the vertex in the directions of the edges...
  
  
• Rational trigonometry
  Rational trigonometry
  Rational trigonometry is a recently introduced approach to trigonometry that eschews all transcendental functions and all proportional measurements of angles. In place of angles, it characterizes the separation between lines by a quantity called the "spread", which is a rational function of their...
  
  
• Spread polynomials
  Spread polynomials
  In the conventional language of trigonometry, the nth-degree spread polynomial Sn, for n = 0, 1, 2, ..., may be characterized by the trigonometric identity\sin^2 = S_n.\,...
  
  

Numerical mathematics

• Abbe error
  Abbe error
  Abbe error, named after Ernst Abbe, also called sine error, describes the magnification of angular error over distance. For example, when one measures a point that is 1 meter away at 45 degrees, an angular error of 1 degree corresponds to a positional error of over 1.745 cm, equivalent to a...
  
  
• Hypot
  Hypot
  Hypot is a mathematical function defined to calculate the length of the hypotenuse of a right-angle triangle. It was designed to avoid errors arising due to limited-precision calculations performed on computers.-Motivation and usage:...
  
  
• Prosthaphaeresis
  Prosthaphaeresis
  Prosthaphaeresis was an algorithm used in the late 16th century and early 17th century for approximate multiplication and division using formulas from trigonometry. For the 25 years preceding the invention of the logarithm in 1614, it was the only known generally-applicable way of approximating...
  
  
• Trigonometric interpolation
  Trigonometric interpolation
  In mathematics, trigonometric interpolation is interpolation with trigonometric polynomials. Interpolation is the process of finding a function which goes through some given data points. For trigonometric interpolation, this function has to be a trigonometric polynomial, that is, a sum of sines and...
  
  

Trigonometric tables

• Generating trigonometric tables
  Generating trigonometric tables
  In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering...
  
  
• Āryabhaṭa's sine table
  Āryabhaṭa's sine table
  Āryabhaṭa's sine table is a set of twenty-four of numbers given in the astronomical treatise Āryabhaṭiya composed by the fifth century Indian mathematician and astronomer Āryabhaṭa , for the computation of the half-chords of certain set of arcs of a circle...
  
  
• Bhaskara I's sine approximation formula
  Bhaskara I's sine approximation formula
  In mathematics, a certain rational expression in one variable for the computation of the approximate values of the trigonometric sines discovered by Bhaskara I , a seventh century Indian mathematician, is known as Bhaskara I's sine approximation formula.This formula is given in his treatise...
  
  
• Madhava's sine table
  Madhava's sine table
  Madhava's sine table is the table of trigonometric sines of various angles constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama. The table lists the trigonometric sines of the twenty-four angles 3.75°, 7.50°, 11.25°, ... , and 90.00°...
  
  
• Ptolemy's table of chords
  Ptolemy's table of chords
  The table of chords, created by the astronomer and geometer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's Almagest, a treatise on mathematical astronomy. It is essentially equivalent to a table of values of the sine function...
  
  
• Rule of marteloio
  Rule of marteloio
  thumb|300px|The tondo e quadro from [[Andrea Bianco]]'s 1436 atlasThe rule of marteloio is a medieval technique of navigational computation that uses compass direction, distance and a simple trigonometric table known as the toleta de marteloio...
  
  

Spherical trigonometry

• Half-side formula
  Half-side formula
  In spherical trigonometry, the half side formula relates the angles and lengths of the sides of spherical triangles, which are triangles drawn on the surface of a sphere and so have curved sides and do not obey the formulas for plane triangles....
  
  
• Haversine formula
  Haversine formula
  The haversine formula is an equation important in navigation, giving great-circle distances between two points on a sphere from their longitudes and latitudes...
  
  

Elementary calculus

• List of integrals of trigonometric functions
• List of integrals of inverse trigonometric functions
• Regiomontanus' angle maximization problem
  Regiomontanus' angle maximization problem
  In mathematics, the Regiomontanus' angle maximization problem, is a famous optimization problem posed by the 15th-century German mathematician Johannes Müller...
  
  
• Trigonometric substitution
  Trigonometric substitution
  In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities to simplify certain integrals containing radical expressions:...
  
  
• Weierstrass substitution
  Weierstrass substitution
  In integral calculus, the Weierstrass substitution, named after Karl Weierstrass, is used for finding antiderivatives, and hence definite integrals, of rational functions of trigonometric functions. No generality is lost by taking these to be rational functions of the sine and cosine. The...
  
  

Other areas of mathematics

• For examples of trigonometric functions as generating functions in combinatorics
  Combinatorics
  Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
  
  , see Alternating permutation.
• Dirichlet kernel
• Euler's formula
  Euler's formula
  Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
  
  
• Exact trigonometric constants
  Exact trigonometric constants
  Exact constant expressions for trigonometric values are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification....
  
  
• Exponential sum
  Exponential sum
  In mathematics, an exponential sum may be a finite Fourier series , or other finite sum formed using the exponential function, usually expressed by means of the functione = \exp.\,...
  
  
• Trigonometric integral
  Trigonometric integral
  In mathematics, the trigonometric integrals are a family of integrals which involve trigonometric functions. A number of the basic trigonometric integrals are discussed at the list of integrals of trigonometric functions.-Sine integral:...
  
  
• Trigonometric number
  Trigonometric number
  In mathematics, a trigonometric number is an irrational number produced by taking the cosine of a rational multiple of a circle, or equivalently, the cosine in radians of a rational multiple of π, or the cosine of a rational number of degrees....
  
  
• Trigonometric polynomial
  Trigonometric polynomial
  In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin and cos with n a natural number. The coefficients may be taken as real numbers, for real-valued functions...
  
  
• Trigonometric series
• Trigonometry in Galois fields
  Trigonometry in Galois fields
  In mathematics, trigonometry analogies are supported by the theory of quadratic extensions of finite fields, also known as Galois fields. The main motivation to deal with a finite field trigonometry is the power of the discrete transforms, which play an important role in engineering and mathematics...
  
  

Geography, geodesy, and land surveying

• Hansen's problem
  Hansen's problem
  Hansen's problem is a problem in planar surveying, named after the astronomer Peter Andreas Hansen , who worked on the geodetic survey of Denmark. There are two known points A and B, and two unknown points P1 and P2. From P1 and P2 an observer measures the angles made by the lines of sight to each...
  
  
• Snellius–Pothenot problem
  Snellius–Pothenot problem
  The Snellius–Pothenot problem is a problem in planar surveying. Given three known points A, B and C an observer at an unknown point P observes that the segment AC subtends an angle \alpha and the segment CB subtends an angle \beta; the problem is to determine the position of the point P...
  
  
• Great-circle distance
  Great-circle distance
  The great-circle distance or orthodromic distance is the shortest distance between any two points on the surface of a sphere measured along a path on the surface of the sphere . Because spherical geometry is rather different from ordinary Euclidean geometry, the equations for distance take on a...
  
   – how to find that distance if one knows the latitude
  Latitude
  In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
  
   and longitude
  Longitude
  Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....
  
  .
• Resection (orientation)
  Resection (orientation)
  Resection is a method for determining a position using a compass and topographic map .-Resection versus intersection:...
  
  
• Vincenty's formulae
  Vincenty's formulae
  Vincenty's formulae are two related iterative methods used in geodesy to calculate the distance between two points on the surface of a spheroid, developed by Thaddeus Vincenty They are based on the assumption that the figure of the Earth is an oblate spheroid, and hence are more accurate than...
  
  
• Geographic distance

History

• Aristarchus' inequality
  Aristarchus' inequality
  In trigonometry, Aristarchus' inequality, named after the ancient astronomer Aristarchus of Samos, states that if α and β are acute angles and β ...
  
  
• Āryabhaṭa's sine table
  Āryabhaṭa's sine table
  Āryabhaṭa's sine table is a set of twenty-four of numbers given in the astronomical treatise Āryabhaṭiya composed by the fifth century Indian mathematician and astronomer Āryabhaṭa , for the computation of the half-chords of certain set of arcs of a circle...
  
  
• Bhaskara I's sine approximation formula
  Bhaskara I's sine approximation formula
  In mathematics, a certain rational expression in one variable for the computation of the approximate values of the trigonometric sines discovered by Bhaskara I , a seventh century Indian mathematician, is known as Bhaskara I's sine approximation formula.This formula is given in his treatise...
  
  
• Jyā, koti-jyā and utkrama-jyā
  Jyā, koti-jyā and utkrama-jyā
  Jyā, koti-jyā and utkrama-jyā are three trigonometric functions introduced by Indian astronomers and mathematicians. The earliest known Indian treatise containing references to these functions is Surya Siddhanta. These are functions of arcs of circles and not functions of angles. Jyā and...
  
  
• Madhava's sine table
  Madhava's sine table
  Madhava's sine table is the table of trigonometric sines of various angles constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama. The table lists the trigonometric sines of the twenty-four angles 3.75°, 7.50°, 11.25°, ... , and 90.00°...
  
  
• Ptolemy's table of chords
  Ptolemy's table of chords
  The table of chords, created by the astronomer and geometer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's Almagest, a treatise on mathematical astronomy. It is essentially equivalent to a table of values of the sine function...
  
  
• Rule of marteloio
  Rule of marteloio
  thumb|300px|The tondo e quadro from [[Andrea Bianco]]'s 1436 atlasThe rule of marteloio is a medieval technique of navigational computation that uses compass direction, distance and a simple trigonometric table known as the toleta de marteloio...
  
  

People

• Aristarchus
  Aristarchus
  -People:* Aristarchus of Samos , Greek astronomer and mathematician* Aristarchus of Samothrace , Greek grammarian* Aristarchus of Tegea , Greek writer* Aristarchus of Thessalonica , Eastern saint...
  
  
• Ptolemy
  Ptolemy
  Claudius Ptolemy , was a 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...
  
  
• Bhaskara I
  Bhaskara I
  Bhāskara was a 7th century Indian mathematician, who was apparently the first to write numbers in the Hindu-Arabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work...
  
  
• Madhava
  Madhava
  Mādhava may be*a Sanskrit patronymic, "descendant of Madhu ".** especially of Krishna or Parashurama as incarnations of Vishnu, see Madhava *** an icon of Krishna...
  
  
• Pythagoras
  Pythagoras
  Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
  
  
• Regiomontanus
  Regiomontanus
  Johannes Müller von Königsberg , today best known by his Latin toponym Regiomontanus, was a German mathematician, astronomer, astrologer, translator and instrument maker....
  
  

External links

• Dave's short trig course
• An introduction to trigonometry
• Trigonometry FAQ
• Trigonometric Delights, by Eli Maor
  Eli Maor
  Eli Maor, an Israel-born historian of mathematics, is the author of several books about the history of mathematics. Eli Maor received his PhD at the Technion – Israel Institute of Technology. He teaches the history of mathematics at Loyola University Chicago...
  
  , 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