Sin

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

sin(((n π)/m)) can be expressed using only square roots if n∈ℤ and m is a product of a power of 2 and distinct Fermat primes {3, 5, 17, 257, …}.

Values at infinities

General characteristics

Domain and analyticity

sin(z) is an entire analytical function of z which is defined over the whole complex z-plane.

Symmetries and periodicities

Parity

sin(z) is an odd function.

Mirror symmetry

Periodicity

sin(z) is a periodic function with period 2π.

Poles and essential singularities

The function sin(z) has only one singular point at z = (OverTilde[∞]).  It is an essential singular point.

Branch points

The function sin(z) does not have branch points.

Branch cuts

The function sin(z) does not have branch cuts.

Series representations

Generalized power series

Expansions at z==(Subscript[z, 0])

Expansions at z==0

Expansions at z==(π/2)

q-series

Exponential Fourier series

Asymptotic series expansions

Other series representations

Residue representations

Dual Taylor series representations

Integral representations

On the real axis

Of the direct function

Contour integral representations

Product representations

Limit representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Ordinary nonlinear differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Argument involving inverse trigonometric and hyperbolic functions

Argument involving complex components

Addition formulas

Half-angle formulas

Multiple arguments

Argument involving numeric multiples of variable

Argument involving symbolic multiples of variable

Some functions of arguments

Products, sums, and powers of the direct function

Products of the direct function

Products involving the direct function

Sums of the direct function

Sums involving the direct function

Powers of the direct function

Powers involving the direct function

Sums of powers involving the direct function

Related transformations

Identities

Functional identities

Univariate functional identities

Biivariate functional identities

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Involving functions of the direct function

Involving functions of the direct function and a power function

Involving functions of the direct function and algebraic functions

Involving functions of the direct function and exponential function

Involving functions of the direct function, exponential and a power functions

Involving functions of the direct function, exponential and algebraic functions

Definite integration

For the direct function itself

Involving the direct function

Dirichlet lemma

Dirichlet lemma

Dirichlet lemma

Dirichlet lemma

Involving related functions

Integral transforms

Fourier exp transforms

Inverse Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Mellin transforms

Hankel transforms

Hilbert transforms

Summation

Finite summation

Infinite summation

Above relation holds in a distributional sense for x∈ℝ.

Above relation holds in a distributional sense for x∈ℝ.

Above relation holds in a distributional sense for x∈ℝ.

Above relation holds in a distributional sense for x∈ℝ.

Products

Finite products

Infinite products

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric0F1

Involving HypergeometricPFQ

Brychkov Yu.A. (2005)

Through Meijer G

Classical cases for the direct function itself

Classical cases for powers of Sin

Classical cases involving Exp

Classical cases involving Sinh

Classical cases involving Cosh

Classical cases involving Cos,CosIntegral,SinIntegral

Classical cases involving ArcCos in the arguments and unit step UnitStep

Classical cases involving ArcTan in the arguments

Classical cases involving ArcCot in the arguments

Classical cases involving ArcSec in the arguments and unit step UnitStep

Classical cases involving Bessel BesselJ

Classical cases involving Cos and Bessel BesselJ

Classical cases involving Bessel BesselY

Classical cases involving Cos and Bessel BesselJ,BesselY

Classical cases involving Hypergeometric0F1

Classical cases involving Hypergeometric0F1Regularized

Generalized cases for the direct function itself

Generalized cases for powers of Sin

Generalized cases involving Exp

Generalized cases involving Cosh

Generalized cases involving Sinh

Generalized cases involving Cos,CosIntegral,SinIntegral

Generalized cases involving Cos and Fresnel FresnelC,FresnelS

Generalized cases involving ArcCos in the arguments and unit step UnitStep

Generalized cases involving ArcTan in the arguments

Generalized cases involving ArcCot in the arguments

Generalized cases involving ArcSec in the arguments and unit step UnitStep

Generalized cases involving Bessel BesselJ

Generalized cases involving Cos and Bessel BesselJ

Generalized cases involving Bessel BesselY

Generalized cases involving Cos and Bessel BesselJ,BesselY

Generalized cases involving Hypergeometric0F1

Generalized cases involving Hypergeometric0F1Regularized

Through other functions

Involving Bessel functions

Involving Jacobi functions

Involving Mathieu functions

Involving some elliptic-type functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With inverse function

With related functions

Involving Exp

Involving Cos

Involving Tan

Involving Cot

Involving Csc

Involving Sec

Involving Sinh

Involving Cosh

Involving Tanh

Involving Coth

Involving Csch

Involving Sech

Inequalities

Zeros

Theorems

Fourier sin transforms

(OverHat[f])(y)==(∫_0^∞)f(x)sin(x y)dx⇔ f(x)==(2/π)(∫_0^∞)(OverHat[f])(y)sin(x y)dy.

Fourier series

f(x)∼((Subscript[a, 0])/2)+(∑_(k=1)^∞)(Subscript[a, k]) cos(((π k x)/r))+(Subscript[b, k])sin(((π k x)/r))/;(Subscript[a, k])==(1/r)(∫_(-r)^r)f(t) cos(((π k t)/r))dt∧(Subscript[b, k])==(1/r)(∫_(-r)^r)f(t) sin(((π k t)/r))dt

De Moivre's theorem

(((cos(z)+i sin(z)))^n)==cos(n z)+i sin(n z)/;   n∈ℕ

The law of sines

For a triangle in the Euclidean plane with edges a ,  b,  c and opposite angles α, β, γ,  the following holds:

((sin(α))/a)==((sin(β))/b)==((sin(γ))/c).

Logistic equation solution

(Subscript[y, n])==(sin^2)((c (2^n)mod 1)π) is a closed form for the chaos-exhibiting logistic equation (Subscript[y, n+1])==4(Subscript[y, n])(1-(Subscript[y, n])).

Weierstrass-Riemann function Ω(x)

Weierstrass-Riemann function Ω(x) , which is continuous everywhere but not differentiable at a dense set of points, has the following representation:

Ω(x)==(∑_(k=1)^∞)((sin((k^2) π x))/((k^2) π))==(π/(4(q^2)))(∑_(k=1)^(q-1))sin((((k^2) p π)/q))/(sin^2)(((k π)/(2 q)))/;x==(p/q)∧p,q∈ℕ∧gcd(p,q)==1.

Coriolis force

The closed form solution of the vector equation (((∂^2) r(t))/(∂(t^2)))==g-2ω×((∂ r(t))/(∂t)) , describing the movement of a mass point  r(t) under the gravitational force  m g and the Coriolis force 2m ω×((∂ r(t))/(∂t))and initial conditions r(t)==(Subscript[r, 0  ])and ∂r(t)/∂t|t==0==(Subscript[v, 0]),  is given by

r(t)==(Subscript[r, 0])+(Subscript[v, 0])t+g ((t^2)/2)-ω×(((2(sin^2)(ω t))/(2(ω^2)))(Subscript[v, 0])+((2ω t-sin(2ω t))/(4(ω^3)))g)+   
                                            ω×(ω×(((2ω t-sin(2ω t))/(2(ω^3)))(Subscript[v, 0])+((2(ω^2) (t^2)-2(sin^2)(ω t))/(4(ω^4)))g)).

Other information

Value properties

History

– K. Ptolemaeus
– Brahmagupta (c. 630)
– R.de Chesters (1140) translated Abu Ja'far Muhammed ibn Musa al-Khwarizme's works and used the word "sine" (in Latin, "sinus"; al-Khwarisme's name is the source of the word "algorithm")
– J. M. Regiomontanus (1464) and A. Dürer (1525)
– F. Maurolyco (1555) used the notation "sinus"
– E. Gunter (1624) used the notation "sin"
– I. Newton (1665) found a series expansion for sin
– L. Euler (1734) found a product representation
– T. Simpson (1750)

The function sin is encountered often in mathematics and the natural sciences.

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