"Sine" and "Cosine" redirect here. For other uses, see Sine (disambiguation) and Cosine (disambiguation). "Sine" is not to be confused with Sign, Sign (mathematics) or the sign function.
Sine and cosine
General information
General definition
Fields of application
Trigonometry, Fourier series, etc.
Trigonometry
Outline
History
Usage
Functions (sin, cos, tan, inverse)
Generalized trigonometry
Reference
Identities
Exact constants
Tables
Unit circle
Laws and theorems
Sines
Cosines
Tangents
Cotangents
Pythagorean theorem
Calculus
Trigonometric substitution
Integrals (inverse functions)
Derivatives
Trigonometric series
Mathematicians
Hipparchus
Ptolemy
Brahmagupta
al-Hasib
al-Battani
Regiomontanus
Viète
de Moivre
Euler
Fourier
v
t
e
In 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 as and .[1]
The definitions of sine and cosine have been extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.
The sine and cosine functions are commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. They can be traced to the jyā and koṭi-jyā functions used in Indian astronomy during the Gupta period.
^Weisstein, Eric W. "Sine". mathworld.wolfram.com. Retrieved 2020-08-29.
In mathematics, sineandcosine are trigonometric functions of an angle. The sineandcosine of an acute angle are defined in the context of a right triangle:...
In mathematics, the Fourier sineandcosine transforms are forms of the Fourier transform that do not use complex numbers or require negative frequency...
mathematics are the sine, the cosine, and the tangent functions. Their reciprocals are respectively the cosecant, the secant, and the cotangent functions...
sine wave of arbitrary phase can be written as the linear combination of two sine waves with phases of zero and a quarter cycle, the sineandcosine components...
In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sineandcosine series are two mathematical series named after Joseph...
expansions for the sine, cosine, and arctangent functions discovered in 14th or 15th century in Kerala, India by the mathematician and astronomer Madhava...
fluid dynamics, and special relativity. The basic hyperbolic functions are: hyperbolic sine "sinh" (/ˈsɪŋ, ˈsɪntʃ, ˈʃaɪn/), hyperbolic cosine "cosh" (/ˈkɒʃ...
trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles...
of the sine function is written sin′(a) = cos(a), meaning that the rate of change of sin(x) at a particular angle x = a is given by the cosine of that...
ordinary sine integral by Si ( i x ) = i Shi ( x ) . {\displaystyle \operatorname {Si} (ix)=i\operatorname {Shi} (x).} The hyperbolic cosine integral...
trigonometric function, and then simplifying the resulting integral with a trigonometric identity. The basic relationship between the sineandcosine is given by...
Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others. The lemniscate sineand lemniscate cosine functions, usually written...
trigonometric series are Fourier series. By expressing a function as a sum of sinesandcosines, many problems involving the function become easier to analyze because...
sum-of-angles formulae, it is one of the basic relations between the sineandcosine functions. The identity is sin 2 θ + cos 2 θ = 1. {\displaystyle...
versus (flipped sine), versinus, versus, or sagitta (arrow). Expressed in terms of common trigonometric functions sine, cosine, and tangent, the versine...
Trigonometric functions. Other definitions, and therefore other proofs are based on the Taylor series of sineandcosine, or on the differential equation f ″...
and μέτρον metron, "measure". The modern words "sine" and "cosine" are derived from the Latin word sinus via mistranslation from Arabic (see Sineand...
functions, the Fourier cosine transform, a, and the Fourier sine transform, b. The function f can be recovered from the sineandcosine transform using f (...
remember trigonometric identities and the relationships between the various trigonometric functions. The sine, cosine, and tangent ratios in a right triangle...
0, 1, −1, 1/2, and −1/2. An angle can be constructed with a compass and straightedge if and only if its sine (or equivalently cosine) can be expressed...
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...
|\sin ax-\cos ax|+C} An integral that is a rational function of the sineandcosine can be evaluated using Bioche's rules. ∫ d x cos a x ± sin a x...
trigonometric functions cosineandsine of angle θ may be defined on the unit circle as follows: If (x, y) is a point on the unit circle, and if the ray from...
Global Information
![{\displaystyle {\begin{aligned}&\sin(\alpha )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}\\[8pt]&\cos(\alpha )={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}\\[8pt]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea2762f231f5fdc1dcfacd59c303106f596ab2e1)
