Misplaced Pages

#95904

48-441: Centigon may refer to: Centigon (unit) , a unit of plane angle, the hundredth part of a gon (gradian). Centigon (company) , a company specializing in security products like armored vehicles Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Centigon . If an internal link led you here, you may wish to change

96-417: A corresponding inverse function , and an analog among the hyperbolic functions . The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles . To extend the sine and cosine functions to functions whose domain is the whole real line , geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then

144-481: A general introduction were made, the unit was only adopted in some countries, and for specialised areas such as surveying , mining and geology . Today, the degree, ⁠ 1 / 360 ⁠ of a turn , or the mathematically more convenient radian, ⁠ 1 / 2 π ⁠ of a turn (used in the SI system of units) is generally used instead. In the 1970s – 1990s, most scientific calculators offered

192-714: A lesser extent in mining and geology . The gon is officially a legal unit of measurement in the European Union and in Switzerland . However, the gradian is not part of the International System of Units (SI). The unit originated in France in connection with the French Revolution as the grade , along with the metric system , hence it is occasionally referred to as a metric degree . Due to confusion with

240-445: A rotation by an angle π {\displaystyle \pi } , the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of π {\displaystyle \pi } . That is, the equalities hold for any angle θ and any integer k . The algebraic expressions for the most important angles are as follows: Writing

288-595: A superscript after the symbol of the function denotes exponentiation , not function composition . For example sin 2 ⁡ x {\displaystyle \sin ^{2}x} and sin 2 ⁡ ( x ) {\displaystyle \sin ^{2}(x)} denote sin ⁡ ( x ) ⋅ sin ⁡ ( x ) , {\displaystyle \sin(x)\cdot \sin(x),} not sin ⁡ ( sin ⁡ x ) . {\displaystyle \sin(\sin x).} This differs from

336-514: Is not part of the International System of Units (SI). The EU directive on the units of measurement notes that the gradian "does not appear in the lists drawn up by the CGPM , CIPM or BIPM ." The most recent, 9th edition of the SI Brochure does not mention the gradian at all. The previous edition mentioned it only in the following footnote: The gon (or grad, where grad is an alternative name for

384-670: Is 360° (particularly in elementary mathematics ). However, in calculus and mathematical analysis , the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers , rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function , via power series, or as solutions to differential equations given particular initial values ( see below ), without reference to any geometric notions. The other four trigonometric functions ( tan , cot , sec , csc ) can be defined as quotients and reciprocals of sin and cos , except where zero occurs in

432-491: Is an arbitrary integer. Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence . Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets. More precisely, defining one has the following series expansions: The following continued fractions are valid in

480-414: Is assigned a range of 100 gon, which eases recognition of the four quadrants, as well as arithmetic involving perpendicular or opposite angles. One advantage of this unit is that right angles to a given angle are easily determined. If one is sighting down a compass course of 117 gon, the direction to one's left is 17 gon, to one's right 217 gon, and behind one 317 gon. A disadvantage

528-408: Is functions that are holomorphic in the whole complex plane, except some isolated points called poles . Here, the poles are the numbers of the form ( 2 k + 1 ) π 2 {\textstyle (2k+1){\frac {\pi }{2}}} for the tangent and the secant, or k π {\displaystyle k\pi } for the cotangent and the cosecant, where k

SECTION 10

#1733106209096

576-461: Is rarely used. Trigonometric functions In 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

624-608: Is that the common angles of 30° and 60° in geometry must be expressed in fractions (as ⁠33 + 1 / 3 ⁠  gon and ⁠66 + 2 / 3 ⁠  gon respectively). In the 18th century, the metre was defined as the 10-millionth part of a quarter meridian . Thus, 1 gon corresponds to an arc length along the Earth's surface of approximately 100 kilometres; 1 centigon to 1 kilometre; 10 microgons to 1 metre. (The metre has been redefined with increasing precision since then.) The gradian

672-501: Is the unique solution with y (0) = 0 . The basic trigonometric functions can be defined by the following power series expansions . These series are also known as the Taylor series or Maclaurin series of these trigonometric functions: The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on

720-403: Is then a definition of the real number π {\displaystyle \pi } which is independent of geometry. Applying the quotient rule to the tangent tan ⁡ x = sin ⁡ x / cos ⁡ x {\displaystyle \tan x=\sin x/\cos x} , so the tangent function satisfies the ordinary differential equation It

768-434: Is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that 2 π {\displaystyle 2\pi } is the smallest value for which they are periodic (i.e., 2 π {\displaystyle 2\pi } is the fundamental period of these functions). However, after

816-454: Is used as its symbol in formulas. Today, the most common versions of these abbreviations are "sin" for sine, "cos" for cosine, "tan" or "tg" for tangent, "sec" for secant, "csc" or "cosec" for cosecant, and "cot" or "ctg" for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as

864-438: The n th term lead to absolutely convergent series: Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions: The following infinite product for the sine is due to Leonhard Euler , and is of great importance in complex analysis: This may be obtained from the partial fraction decomposition of cot ⁡ z {\displaystyle \cot z} given above, which

912-459: The Euclidean plane that are related to the unit circle , which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between 0 and π 2 {\textstyle {\frac {\pi }{2}}} radians (90°), the unit circle definitions allow

960-559: The Pythagorean identity . The other trigonometric functions can be found along the unit circle as By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is Since a rotation of an angle of ± 2 π {\displaystyle \pm 2\pi } does not change

1008-709: The function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation , for example sin( x ) . Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression sin ⁡ x + y {\displaystyle \sin x+y} would typically be interpreted to mean sin ⁡ ( x ) + y , {\displaystyle \sin(x)+y,} so parentheses are required to express sin ⁡ ( x + y ) . {\displaystyle \sin(x+y).} A positive integer appearing as

SECTION 20

#1733106209096

1056-806: The reciprocal . For example sin − 1 ⁡ x {\displaystyle \sin ^{-1}x} and sin − 1 ⁡ ( x ) {\displaystyle \sin ^{-1}(x)} denote the inverse trigonometric function alternatively written arcsin ⁡ x : {\displaystyle \arcsin x\colon } The equation θ = sin − 1 ⁡ x {\displaystyle \theta =\sin ^{-1}x} implies sin ⁡ θ = x , {\displaystyle \sin \theta =x,} not θ ⋅ sin ⁡ x = 1. {\displaystyle \theta \cdot \sin x=1.} In this case,

1104-408: The (historically later) general functional notation in which f 2 ( x ) = ( f ∘ f ) ( x ) = f ( f ( x ) ) . {\displaystyle f^{2}(x)=(f\circ f)(x)=f(f(x)).} However, the exponent − 1 {\displaystyle {-1}} is commonly used to denote the inverse function , not

1152-408: The angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete turn (360°) is an angle of 2 π (≈ 6.28) rad. For real number x , the notation sin x , cos x , etc. refers to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended,

1200-455: The definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number. Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry. Various ways exist in the literature for defining the trigonometric functions in a manner suitable for analysis; they include: Sine and cosine can be defined as

1248-458: The degree sign must be explicitly shown ( sin x° , cos x° , etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180 x / π )°, so that, for example, sin π = sin 180° when we take x = π . In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π /180 ≈ 0.0175. The six trigonometric functions can be defined as coordinate values of points on

1296-487: The denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle in radians. Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions. Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures. When radians (rad) are employed,

1344-474: The domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations . This allows extending the domain of sine and cosine functions to the whole complex plane , and the domain of the other trigonometric functions to the complex plane with some isolated points removed. Conventionally, an abbreviation of each trigonometric function's name

1392-498: The domain of trigonometric functions to be extended to all positive and negative real numbers. Let L {\displaystyle {\mathcal {L}}} be the ray obtained by rotating by an angle θ the positive half of the x -axis ( counterclockwise rotation for θ > 0 , {\displaystyle \theta >0,} and clockwise rotation for θ < 0 {\displaystyle \theta <0} ). This ray intersects

1440-665: The existing term grad(e) in some northern European countries (meaning a standard degree, ⁠ 1 / 360 ⁠ of a turn), the name gon was later adopted, first in those regions, and later as the international standard. In France, it was also called grade nouveau . In German , the unit was formerly also called Neugrad (new degree) (whereas the standard degree was referred to as Altgrad (old degree)), likewise nygrad in Danish , Swedish and Norwegian (also gradian ), and nýgráða in Icelandic . Although attempts at

1488-404: The gon) is an alternative unit of plane angle to the degree, defined as (π/200) rad. Thus there are 100 gon in a right angle. The potential value of the gon in navigation is that because the distance from the pole to the equator of the Earth is approximately 10 000  km , 1 km on the surface of the Earth subtends an angle of one centigon at the centre of the Earth. However the gon

Centigon - Misplaced Pages Continue

1536-640: The gon, as well as radians and degrees, for their trigonometric functions . In the 2010s, some scientific calculators lack support for gradians. The international standard symbol for this unit today is "gon" (see ISO 31-1 ). Other symbols used in the past include "gr", "grd", and "g", the last sometimes written as a superscript, similarly to a degree sign: 50 = 45°. A metric prefix is sometimes used, as in "dgon", "cgon", "mgon", denoting respectively 0.1 gon, 0.01 gon, 0.001 gon. Centesimal arc-minutes and centesimal arc-seconds were also denoted with superscripts and , respectively. Each quadrant

1584-731: The line of equation y = 1 {\displaystyle y=1} at point C = ( x C , 1 ) . {\displaystyle \mathrm {C} =(x_{\mathrm {C} },1).} The tangent line to the unit circle at the point A , is perpendicular to L , {\displaystyle {\mathcal {L}},} and intersects the y - and x -axes at points D = ( 0 , y D ) {\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })} and E = ( x E , 0 ) . {\displaystyle \mathrm {E} =(x_{\mathrm {E} },0).} The coordinates of these points give

1632-512: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Centigon&oldid=932752073 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Centigon (unit) In continental Europe , the French word centigrade , also known as centesimal minute of arc ,

1680-441: The numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values. Such simple expressions generally do not exist for other angles which are rational multiples of a right angle. The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees. G. H. Hardy noted in his 1908 work A Course of Pure Mathematics that

1728-417: The position or size of a shape, the points A , B , C , D , and E are the same for two angles whose difference is an integer multiple of 2 π {\displaystyle 2\pi } . Thus trigonometric functions are periodic functions with period 2 π {\displaystyle 2\pi } . That is, the equalities hold for any angle θ and any integer k . The same

1776-434: The right-angled triangle definition, by taking the right-angled triangle to have the unit radius OA as hypotenuse . And since the equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} holds for all points P = ( x , y ) {\displaystyle \mathrm {P} =(x,y)} on the unit circle, this definition of cosine and sine also satisfies

1824-413: The same ordinary differential equation Sine is the unique solution with y (0) = 0 and y ′(0) = 1 ; cosine is the unique solution with y (0) = 1 and y ′(0) = 0 . One can then prove, as a theorem, that solutions cos , sin {\displaystyle \cos ,\sin } are periodic, having the same period. Writing this period as 2 π {\displaystyle 2\pi }

1872-401: The same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table. In geometric applications, the argument of a trigonometric function is generally the measure of an angle . For this purpose, any angular unit is convenient. One common unit is degrees , in which a right angle is 90° and a complete turn

1920-420: The simplest periodic functions , and as such are also widely used for studying periodic phenomena through Fourier analysis . The trigonometric functions most widely used in modern mathematics are the sine , the cosine , and the tangent functions. Their reciprocals are respectively the cosecant , the secant , and the cotangent functions, which are less used. Each of these six trigonometric functions has

1968-457: The superscript could be considered as denoting a composed or iterated function , but negative superscripts other than − 1 {\displaystyle {-1}} are not in common use. If the acute angle θ is given, then any right triangles that have an angle of θ are similar to each other. This means that the ratio of any two side lengths depends only on θ . Thus these six ratios define six functions of θ , which are

Centigon - Misplaced Pages Continue

2016-740: The trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, opposite represents the side opposite the given angle θ , and adjacent represents the side between the angle θ and the right angle. Various mnemonics can be used to remember these definitions. In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or ⁠ π / 2 ⁠ radians . Therefore sin ⁡ ( θ ) {\displaystyle \sin(\theta )} and cos ⁡ ( 90 ∘ − θ ) {\displaystyle \cos(90^{\circ }-\theta )} represent

2064-685: The unique solution to the initial value problem : Differentiating again, d 2 d x 2 sin ⁡ x = d d x cos ⁡ x = − sin ⁡ x {\textstyle {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x} and d 2 d x 2 cos ⁡ x = − d d x sin ⁡ x = − cos ⁡ x {\textstyle {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x} , so both sine and cosine are solutions of

2112-549: The unit circle at the point A = ( x A , y A ) . {\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).} The ray L , {\displaystyle {\mathcal {L}},} extended to a line if necessary, intersects the line of equation x = 1 {\displaystyle x=1} at point B = ( 1 , y B ) , {\displaystyle \mathrm {B} =(1,y_{\mathrm {B} }),} and

2160-418: The values of all trigonometric functions for any arbitrary real value of θ in the following manner. The trigonometric functions cos and sin are defined, respectively, as the x - and y -coordinate values of point A . That is, In the range 0 ≤ θ ≤ π / 2 {\displaystyle 0\leq \theta \leq \pi /2} , this definition coincides with

2208-415: The whole complex plane . Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation. Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions , that

2256-455: The whole complex plane: The last one was used in the historically first proof that π is irrational . There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match: This identity can be proved with the Herglotz trick. Combining the (– n ) th with

2304-495: Was in use for one hundredth of a grade; similarly, the centesimal second of arc was defined as one hundredth of a centesimal arc-minute, analogous to decimal time and the sexagesimal minutes and seconds of arc . The chance of confusion was one reason for the adoption of the term Celsius to replace centigrade as the name of the temperature scale. Gradians are principally used in surveying (especially in Europe), and to

#95904