--- title: "Sin" language: "en" type: "Symbol" summary: "Sin[z] gives the sine of z." keywords: - sen - sin - sine - sinus - sin - SIN - csin - csinf - csinl - sin - sinf - sinl - sin - sin - sin - sin - sin - sind canonical_url: "https://reference.wolfram.com/language/ref/Sin.html" source: "Wolfram Language Documentation" related_guides: - title: "Trigonometric Functions" link: "https://reference.wolfram.com/language/guide/TrigonometricFunctions.en.md" - title: "GPU Computing" link: "https://reference.wolfram.com/language/guide/GPUComputing.en.md" - title: "Precollege Education" link: "https://reference.wolfram.com/language/guide/PrecollegeEducation.en.md" - title: "GPU Computing with Apple" link: "https://reference.wolfram.com/language/guide/GPUComputing-Apple.en.md" - title: "GPU Computing with NVIDIA" link: "https://reference.wolfram.com/language/guide/GPUComputing-NVIDIA.en.md" - title: "Functions for Separable Coordinate Systems" link: "https://reference.wolfram.com/language/guide/FunctionsForSeparableCoordinateSystems.en.md" - title: "Mathematical Functions" link: "https://reference.wolfram.com/language/guide/MathematicalFunctions.en.md" - title: "Elementary Functions" link: "https://reference.wolfram.com/language/guide/ElementaryFunctions.en.md" related_functions: - title: "AngleVector" link: "https://reference.wolfram.com/language/ref/AngleVector.en.md" - title: "ArcSin" link: "https://reference.wolfram.com/language/ref/ArcSin.en.md" - title: "Cos" link: "https://reference.wolfram.com/language/ref/Cos.en.md" - title: "Tan" link: "https://reference.wolfram.com/language/ref/Tan.en.md" - title: "Csc" link: "https://reference.wolfram.com/language/ref/Csc.en.md" - title: "Degree" link: "https://reference.wolfram.com/language/ref/Degree.en.md" - title: "SinDegrees" link: "https://reference.wolfram.com/language/ref/SinDegrees.en.md" - title: "TrigToExp" link: "https://reference.wolfram.com/language/ref/TrigToExp.en.md" - title: "TrigExpand" link: "https://reference.wolfram.com/language/ref/TrigExpand.en.md" - title: "Sinc" link: "https://reference.wolfram.com/language/ref/Sinc.en.md" - title: "Haversine" link: "https://reference.wolfram.com/language/ref/Haversine.en.md" - title: "CirclePoints" link: "https://reference.wolfram.com/language/ref/CirclePoints.en.md" related_tutorials: - title: "Some Mathematical Functions" link: "https://reference.wolfram.com/language/tutorial/SomeMathematicalFunctions.en.md" - title: "Elementary Transcendental Functions" link: "https://reference.wolfram.com/language/tutorial/MathematicalFunctions.en.md#10797" --- # Sin Sin[z] gives the sine of z. ## Details * Mathematical function, suitable for both symbolic and numerical manipulation. * Unless explicitly given as a ``Quantity`` object, the argument of ``Sin`` is assumed to be in radians. (Multiply by ``Degree`` to convert from degrees.) » * ``Sin`` is automatically evaluated when its argument is a simple rational multiple of $\pi$; for more complicated rational multiples, ``FunctionExpand`` can sometimes be used. » * For certain special arguments, ``Sin`` automatically evaluates to exact values. * ``Sin`` can be evaluated to arbitrary numerical precision. * ``Sin`` automatically threads over lists. » * ``Sin`` can be used with ``Interval`` and ``CenteredInterval`` objects. » --- ## Background & Context ``Sin`` is the sine function, which is one of the basic functions encountered in trigonometry. It is defined for real numbers by letting $x$ be a radian angle measured counterclockwise from the $x$ axis along the circumference of the unit circle. ``Sin[x]`` then gives the vertical coordinate of the arc endpoint. The equivalent schoolbook definition of the sine of an angle $\theta$ in a right triangle is the ratio of the length of the leg opposite $\theta$ to the length of the hypotenuse. ``Sin`` automatically evaluates to exact values when its argument is a simple rational multiple of $\pi$. For more complicated rational multiples, ``FunctionExpand`` can sometimes be used to obtain an explicit exact value. To specify an argument using an angle measured in degrees, the symbol ``Degree`` can be used as a multiplier (e.g. ``Sin[30 Degree]``). When given exact numeric expressions as arguments, ``Sin`` may be evaluated to arbitrary numeric precision. Other operations useful for manipulation of symbolic expressions involving ``Sin`` include ``TrigToExp``, ``TrigExpand``, ``Simplify``, and ``FullSimplify``. ``Sin`` threads element-wise over lists and matrices. In contrast, ``MatrixFunction`` can be used to give the sine of a square matrix (i.e. the power series for the sine function with ordinary powers replaced by matrix powers). ``Sin`` is periodic with period $2\pi$, as reported by ``FunctionPeriod``. ``Sin`` satisfies the identity $\sin ^2(x)+\cos ^2(x)=1$, which is equivalent to the Pythagorean theorem. The definition of the sine function is extended to complex arguments $z$ using the definition $\sin (z)=\frac{1}{2} i\left(e^{-i z}-e^{i z}\right)$, where $e$ is the base of the natural logarithm. The sine function is entire, meaning it is complex differentiable at all finite points of the complex plane. ``Sin[z]`` has series expansion $\sum _{k=0}^{\infty } \frac{(-1)^{k-1}}{(2 k-1)!}z^{2 k-1}$ about the origin. The inverse function of ``Sin`` is ``ArcSin``. The hyperbolic sine is given by ``Sinh``. Other related mathematical functions include ``Cos``, ``Tan``, and ``Csc``. --- ## Examples (96) ### Basic Examples (5) The argument is given in radians: ```wl In[1]:= Sin[Pi / 3] Out[1]= (Sqrt[3]/2) ``` --- Use ``Degree`` to specify an argument in degrees: ```wl In[1]:= Sin[60Degree] Out[1]= (Sqrt[3]/2) ``` --- Plot over a subset of the reals: ```wl In[1]:= Plot[Sin[x], {x, 0, 2Pi}] Out[1]= [image] ``` --- Plot over a subset of the complexes: ```wl In[1]:= ComplexPlot3D[Sin[z], {z, -2 π - 2 I, 2 π + 2 I}, PlotLegends -> Automatic] Out[1]= [image] ``` --- Series expansion at ``0`` : ```wl In[1]:= Series[Sin[x], {x, 0, 10}] Out[1]= SeriesData[x, 0, {1, 0, Rational[-1, 6], 0, Rational[1, 120], 0, Rational[-1, 5040], 0, Rational[1, 362880]}, 1, 11, 1] ``` ### Scope (52) #### Numerical Evaluation (6) Evaluate numerically: ```wl In[1]:= Sin[1.2] Out[1]= 0.932039 ``` --- Evaluate to high precision: ```wl In[1]:= N[Sin[12 / 10], 50] Out[1]= 0.93203908596722634967013443549482599541507058820873 ``` The precision of the output tracks the precision of the input: ```wl In[2]:= Sin[1.20000000000000000000000] Out[2]= 0.93203908596722634967013 ``` --- ``Sin`` can take complex number inputs: ```wl In[1]:= Sin[2.5 + I] Out[1]= 0.923491  - 0.941505 I ``` --- Evaluate ``Sin`` efficiently at high precision: ```wl In[1]:= Sin[1.2`500]//Timing Out[1]= {0., 0.9320390859672263496701344354948259954150705882087307353665978944502423415767920542157417224381184959624520224612545817096897563647461506673942640388109579314371616639738436244547639883546898824108774382629750864159214083332477074169521019125 ... 636880337265330195583341366633371125721553603181389759436200996925490801559848123977367325317750644904632080074678777828952095594520998082787389557388695664075468596880233521917254876361615338190696183311770949513341060729059576423577552810372800} In[2]:= Sin[1.2`100000];//Timing Out[2]= {0.140401, Null} ``` --- Compute the elementwise values of an array using automatic threading: ```wl In[1]:= Sin[ {{5π / 6, 0}, {3π / 2, -π / 2}}] Out[1]= {{(1/2), 0}, {-1, -1}} ``` Or compute the matrix ``Sin`` function using ``MatrixFunction``: ```wl In[2]:= MatrixFunction[Sin[#]&, {{5π / 6, 0}, {3π / 2, -π / 2}}] Out[2]= {{(1/2), 0}, {(27/16), -1}} ``` --- Compute worst-case guaranteed intervals using ``Interval`` and ``CenteredInterval`` objects: ```wl In[1]:= Sin[Interval[{-Pi / 6, Pi / 6}]] Out[1]= Interval[{-(1/2), (1/2)}] In[2]:= Sin[Interval[{-Infinity, Infinity}]] Out[2]= Interval[{-1, 1}] In[3]:= Sin[CenteredInterval[1, 1 / 100]] Out[3]= CenteredInterval[{{947364733195477, -50, 742573588, -37}, 30}] In[4]:= Sin[CenteredInterval[2 + 3I, (1 + I) / 100]] Out[4]= CenteredInterval[{{{1383388796920080164528699, -77, 572965171, -32}, {-10079798521198524426808325, -81, 575120415, -32}}, 30}] ``` Or compute average-case statistical intervals using ``Around``: ```wl In[5]:= Sin[Around[2, 0.01]] Out[5]= Around[0.9092974268256817, 0.004161468365471424] ``` #### Specific Values (6) Values of ``Sin`` at fixed points: ```wl In[1]:= Table[Sin[n (π/6)], {n, 0, 6}] Out[1]= {0, (1/2), (Sqrt[3]/2), 1, (Sqrt[3]/2), (1/2), 0} ``` --- ``Sin`` has exact values at rational multiples of pi: ```wl In[1]:= Table[Sin[n (π/6)], {n, 0, 6}] Out[1]= {0, (1/2), (Sqrt[3]/2), 1, (Sqrt[3]/2), (1/2), 0} ``` --- Values at infinity: ```wl In[1]:= Sin[Infinity] Out[1]= Interval[{-1, 1}] In[2]:= Sin[ComplexInfinity] Out[2]= Indeterminate ``` --- Simple exact values are generated automatically: ```wl In[1]:= Sin[Pi / 5] Out[1]= Sqrt[(5/8) - (Sqrt[5]/8)] ``` More complicated cases require explicit use of ``FunctionExpand`` : ```wl In[2]:= Sin[Pi / 24] Out[2]= Sin[(π/24)] In[3]:= FunctionExpand[%] Out[3]= (1/4) Sqrt[2 - Sqrt[2]] (1 + Sqrt[2]) - (1/4) (-1 + Sqrt[2]) Sqrt[3 (2 + Sqrt[2])] ``` --- Zeros of ``Sin`` : ```wl In[1]:= Assuming[m∈Integers, Refine[Sin[π m]]] Out[1]= 0 ``` --- Extrema of ``Sin`` : ```wl In[1]:= Assuming[m∈Integers, FullSimplify[Sin[π ((1/2) + m)]]] Out[1]= (-1)^m ``` Find the first positive maximum as a root of $\frac{d \sin (x)}{\text{dx}}=0$ : ```wl In[2]:= sol = Solve[D[Sin[x], x] == 0 && 0 < x < π, x] Out[2]= {{x -> (π/2)}} ``` Substitute in the result: ```wl In[3]:= xmax = x /. First[sol] Out[3]= (π/2) ``` Visualize the result: ```wl In[4]:= Plot[Sin[x], {x, 0, 2π}, Epilog -> Style[Point[{xmax, Sin[xmax]}], PointSize[Large], Red]] Out[4]= [image] ``` #### Visualization (3) Plot the ``Sin`` function: ```wl In[1]:= Plot[Sin[x], {x, 0, 4π}] Out[1]= [image] ``` --- Plot the real part of $\sin (z)$ : ```wl In[1]:= ComplexContourPlot[Re[Sin[z]], {z, -2π - 2I, 2π + 2I}, IconizedObject[«PlotOptions»]] Out[1]= [image] ``` Plot the imaginary part of $\sin (z)$ : ```wl In[2]:= ComplexContourPlot[Im[Sin[z]], {z, -2π - 2I, 2π + 2I}, IconizedObject[«PlotOptions»]] Out[2]= [image] ``` --- Polar plot with $r=\sin (k \phi )$ : ```wl In[1]:= Table[PolarPlot[Sin[k ϕ], {ϕ, 0, 2 π}, Sequence[Frame -> True, FrameTicks -> {{{-1, -0.5, 0, 0.5, 1}, None}, {{-1, -0.5, 0, 0.5, 1}, None}}, PlotLabel -> "k=" <> ToString[k]]], {k, 1, 8}] Out[1]= [image] ``` #### Function Properties (13) ``Sin`` is defined for all real and complex values: ```wl In[1]:= FunctionDomain[Sin[x], x] Out[1]= True In[2]:= FunctionDomain[Sin[z], z, Complexes] Out[2]= True ``` --- ``Sin`` achieves all real values between $-1$ and 1: ```wl In[1]:= FunctionRange[Sin[x], x, y] Out[1]= -1 ≤ y ≤ 1 ``` The range for complex values is the whole plane: ```wl In[2]:= FunctionRange[Sin[z], z, y, Complexes] Out[2]= True ``` --- ``Sin`` is a periodic function with a period $2 \pi$ : ```wl In[1]:= FunctionPeriod[Sin[x], x] Out[1]= 2 π ``` --- ``Sin`` is an odd function: ```wl In[1]:= Sin[-x] Out[1]= -Sin[x] ``` --- ``Sin`` has the mirror property $\sin \left(z^*\right)=\sin (z)^*$ : ```wl In[1]:= FullSimplify[Sin[Conjugate[z]] == Conjugate[Sin[z]]] Out[1]= True ``` --- ``Sin`` is an analytic function of ``x`` : ```wl In[1]:= FunctionAnalytic[Sin[x], x] Out[1]= True ``` --- ``Sin`` is monotonic in a specific range: ```wl In[1]:= FunctionMonotonicity[Sin[x], x] Out[1]= Indeterminate In[2]:= FunctionMonotonicity[{Sin[x], 0 < x < π / 2}, x] Out[2]= 1 ``` --- ``Sin`` is not injective: ```wl In[1]:= FunctionInjective[Sin[x], x] Out[1]= False In[2]:= Plot[{Sin[x], 1 / 2}, {x, -2π, 2π}] Out[2]= [image] ``` --- ``Sin`` is not surjective: ```wl In[1]:= FunctionSurjective[Sin[x], x] Out[1]= False In[2]:= Plot[{Sin[x], 1.5}, {x, -2π, 2π}] Out[2]= [image] ``` --- ``Sin`` is neither non-negative nor non-positive: ```wl In[1]:= FunctionSign[Sin[x], x] Out[1]= Indeterminate ``` --- ``Sin`` has no singularities or discontinuities: ```wl In[1]:= FunctionSingularities[Sin[x], x] Out[1]= False In[2]:= FunctionDiscontinuities[Sin[x], x] Out[2]= False ``` --- ``Sin`` is neither convex nor concave: ```wl In[1]:= FunctionConvexity[Sin[x], x] Out[1]= Indeterminate ``` ``Sin`` is concave for ``x`` in ``[0, π]`` : ```wl In[2]:= FunctionConvexity[{Sin[x], 0 < x < π}, x] Out[2]= -1 In[3]:= Plot[Sin[x], {x, 0, π}] Out[3]= [image] ``` --- ``TraditionalForm`` formatting: ```wl In[1]:= Sin[α]//TraditionalForm Out[1]//TraditionalForm= $$\sin (\alpha )$$ ``` #### Differentiation (3) First derivative: ```wl In[1]:= D[Sin[x], x] Out[1]= Cos[x] ``` --- Higher derivatives: ```wl In[1]:= Table[D[Sin[x], {x, n}], {n, 1, 4}] Out[1]= {Cos[x], -Sin[x], -Cos[x], Sin[x]} In[2]:= Plot[Evaluate[%], {x, -2π, 2π}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative", "Fourth Derivative"}] Out[2]= [image] ``` --- Formula for the $n$$$^{\text{th}}$$ derivative: ```wl In[1]:= D[Sin[x], {x, n}] Out[1]= Sin[(n π/2) + x] ``` #### Integration (3) Compute the indefinite integral using ``Integrate`` : ```wl In[1]:= Integrate[Sin[x], x] Out[1]= -Cos[x] ``` --- Definite integral of ``Sin`` over a period is 0: ```wl In[1]:= Integrate[Sin[x], {x, 0, 2 π}] Out[1]= 0 ``` --- More integrals: ```wl In[1]:= Integrate[Sin[x]Cos[x], x] Out[1]= -(1/2) Cos[x]^2 In[2]:= Integrate[Sin[z] ^ a, z] Out[2]= -Cos[z] Hypergeometric2F1[(1/2), (1 - a/2), (3/2), Cos[z]^2] Sin[z]^1 + a (Sin[z]^2)^(1/2) (-1 - a) ``` #### Series Expansions (4) Find the Taylor expansion using ``Series`` : ```wl In[1]:= Series[Sin[x], {x, 0, 7}] Out[1]= SeriesData[x, 0, {1, 0, Rational[-1, 6], 0, Rational[1, 120], 0, Rational[-1, 5040]}, 1, 8, 1] ``` Plots of the first three approximations for ``Sin`` around $x=0$ : ```wl In[2]:= terms = Normal@Table[Series[Sin[x], {x, 0, m}], {m, 1, 5, 2}]; Plot[{Sin[x], terms}, {x, -2π, 2π}, PlotRange -> {-1.5, 1.5}] Out[2]= [image] ``` --- General term in the series expansion using ``SeriesCoefficient`` : ```wl In[1]:= SeriesCoefficient[Sin[x], {x, 0, n}] Out[1]= Piecewise[{{(I*I^n*(-1 + (-1)^n))/(2*n!), n >= 0}}, 0] ``` --- Fourier series: ```wl In[1]:= FourierSeries[Sin[z], z, 1] Out[1]= (1/2) I E^-I z - (1/2) I E^I z ``` --- ``Sin`` can be applied to power series: ```wl In[1]:= Sin[(π/2) + x + (x^2/2) + (x^3/3) + O[x]^4] Out[1]= SeriesData[x, 0, {1, 0, Rational[-1, 2], Rational[-1, 2]}, 0, 4, 1] ``` #### Integral Transforms (3) Compute the Fourier transform using ``FourierTransform`` : ```wl In[1]:= FourierTransform[Sin[t], t, ω ] Out[1]= I Sqrt[(π/2)] DiracDelta[-1 + ω] - I Sqrt[(π/2)] DiracDelta[1 + ω] ``` --- ``LaplaceTransform`` : ```wl In[1]:= LaplaceTransform[Sin[t], t, s ] Out[1]= (1/1 + s^2) ``` --- ``MellinTransform`` : ```wl In[1]:= MellinTransform[Sin[x], x, s ] Out[1]= Gamma[s] Sin[(π s/2)] ``` #### Function Identities and Simplifications (6) Double-angle formula using ``TrigExpand`` : ```wl In[1]:= TrigExpand[Sin[2x]] Out[1]= 2 Cos[x] Sin[x] ``` --- Angle sum formula: ```wl In[1]:= TrigExpand[Sin[x + y]] Out[1]= Cos[y] Sin[x] + Cos[x] Sin[y] ``` --- Multiple‐angle expressions: ```wl In[1]:= TrigExpand[Sin[4x]] Out[1]= 4 Cos[x]^3 Sin[x] - 4 Cos[x] Sin[x]^3 ``` Recover the original expression using ``TrigReduce`` : ```wl In[2]:= TrigReduce[%] Out[2]= Sin[4 x] ``` --- Convert sums to products using ``TrigFactor`` : ```wl In[1]:= TrigFactor[Sin[x] + Sin[y]] Out[1]= 2 Cos[(x/2) - (y/2)] Sin[(x/2) + (y/2)] ``` --- Expand using ``ComplexExpand`` assuming real variables ``x`` and ``y`` : ```wl In[1]:= ComplexExpand[Sin[x + I y]] Out[1]= Cosh[y] Sin[x] + I Cos[x] Sinh[y] ``` --- Convert to exponentials using ``TrigToExp`` : ```wl In[1]:= TrigToExp[Sin[z]] Out[1]= (1/2) I E^-I z - (1/2) I E^I z ``` #### Function Representations (5) Use ``Simplify`` to find a representation through ``Cos`` : ```wl In[1]:= Simplify[Cos[(Pi/2) - x]] Out[1]= Sin[x] ``` --- Representation through Bessel functions: ```wl In[1]:= Simplify[Sqrt[( π x/2)]BesselJ[(1/2), x]] Out[1]= Sin[x] In[2]:= Simplify[-I Sqrt[( π I x/2)]BesselI[(1/2), I x]] Out[2]= Sin[x] ``` --- Representation through ``SphericalHarmonicY`` : ```wl In[1]:= Simplify[Sqrt[8 π / 3]SphericalHarmonicY[1, -1, θ, 0]] Out[1]= Sin[θ] ``` --- Representation in terms of ``MeijerG`` : ```wl In[1]:= MeijerGReduce[Sin[x], x] Out[1]= Sqrt[π] Inactive[MeijerG][{{}, {}}, {{(1/2)}, {0}}, (x/2), (1/2)] In[2]:= Activate[%] Out[2]= Sin[x] ``` --- ``Sin`` can be represented as a ``DifferentialRoot`` : ```wl In[1]:= DifferentialRootReduce[Sin[x], x] Out[1]= DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {\[FormalY][\[FormalX]] + Derivative[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][0] == 0, Derivative[1][\[FormalY]][0] == 1}]][x] ``` ### Applications (15) Draw a circle: ```wl In[1]:= ParametricPlot[{Sin[t], Cos[t]}, {t, 0, 2Pi}] Out[1]= [image] ``` --- Lissajous figure: ```wl In[1]:= ParametricPlot[{Sin[t], Sin[2t]}, {t, 0, 2Pi}] Out[1]= [image] ``` --- Equiangular (logarithmic) spiral: ```wl In[1]:= ParametricPlot[Exp[t / 10]{Sin[t], Cos[t]}, {t, 0, 10Pi}, PlotRange -> All] Out[1]= [image] ``` --- Motion in a circle: ```wl In[1]:= Animate[Graphics[Line[{{0, 0}, {Sin[t], Cos[t]}}], PlotRange -> 1.2], {t, 0, 10Pi}] Out[1]= DynamicModule[«8»] ``` --- Play a pure tone at 440 Hz: ```wl In[1]:= Play[Sin[2 Pi 440 t], {t, 0, 1}] Out[1]= Sound[SampledSoundFunction[CompiledFunction[{10, 11., 5444}, {_Integer}, {{2, 0, 0}, {3, 0, 5}}, {{1., {3, 0, 6}}, {440, {2, 0, 2}}, {0.000125, {3, 0, 1}}, {3.141592653589793, {3, 0, 4}}, {2, {2, 0, 1}}, {0., {3, 0, 0}}}, {0, 3, 7, 0, 0}, ... , {10, 1, 3}, {10, 2, 5}, {16, 3, 4, 5, 2, 3}, {40, 1, 3, 0, 3, 3, 0, 5}, {13, 5, 0, 5}, {16, 5, 6, 5}, {1}}, Function[{Play`Time813}, Block[{t = 0. + 0.000125*Play`Time813}, (Sin[((2*Pi)*440)*t] + 0.)*1.]], Evaluate], 8000, 8000]] ``` --- Solve an equation for harmonic motion: ```wl In[1]:= DSolve[x''[t] + ω^2x[t] == 0, x[t], t] Out[1]= {{x[t] -> C[1] Cos[t ω] + C[2] Sin[t ω]}} ``` --- Rotation matrix: ```wl In[1]:= RotationMatrix[θ] Out[1]= {{Cos[θ], -Sin[θ]}, {Sin[θ], Cos[θ]}} ``` Rotate a vector: ```wl In[2]:= %.{x, y} Out[2]= {x Cos[θ] - y Sin[θ], y Cos[θ] + x Sin[θ]} ``` --- Plot a sphere: ```wl In[1]:= ParametricPlot3D[{Cos[ϕ] Sin[θ], Sin[θ] Sin[ϕ], Cos[θ]}, {ϕ, -π, π}, {θ, 0, π}] Out[1]= [image] ``` --- Plot a torus: ```wl In[1]:= ParametricPlot3D[{Cos[ϕ] + 1 / 2 Cos[θ] Cos[ϕ], Sin[ϕ] + 1 / 2 Cos[θ] Sin[ϕ], Sin[θ] / 2}, {ϕ, -π, π}, {θ, 0, 2 π}] Out[1]= [image] ``` --- Waves: ```wl In[1]:= Plot3D[Sin[x]Sin[y], {x, 0, 10Pi}, {y, 0, 10Pi}] Out[1]= [image] ``` --- Triple‐periodic surface: ```wl In[1]:= ContourPlot3D[Sin[x] + Sin[y] + Sin[z], {x, -2Pi, 2Pi}, {y, -2Pi, 2Pi}, {z, -2Pi, 2Pi}, Contours -> {0}, Mesh -> False, BoundaryStyle -> None, ContourStyle -> Opacity[0.8]] Out[1]= [image] ``` --- Approximate the almost nowhere differentiable Riemann–Weierstrass function: ```wl In[1]:= Plot[Sum[N[Sin[j ^ 2 x] / j ^ 2], {j, 1, 12}], {x, 0, 2Pi}] Out[1]= [image] ``` --- Intensity of the Fraunhofer diffraction pattern of a circular aperture versus diffraction angle: ```wl In[1]:= Plot[2(BesselJ[2, 10Sin[θ]] / (10Sin[θ])) ^ 2, {θ, 0, π / 2}] Out[1]= [image] ``` --- Encode graphics in a QR code: ```wl In[1]:= str = ToString[HoldForm[Plot[Sin[x], {x, 0, 20}]]] Out[1]= "Plot[Sin[x], {x, 0, 20}]" In[2]:= i = BarcodeImage[str, "QR"] Out[2]= [image] ``` Decode and evaluate the expression: ```wl In[3]:= ToExpression[BarcodeRecognize[i]] Out[3]= [image] ``` --- Find a point on a unit circle using ``Cos`` and ``Sin`` functions: ```wl In[1]:= ContourPlot[x ^ 2 + y ^ 2 == 1, {x, -1.5, 1.5}, {y, -1.5, 1.5}, Epilog -> Style[Point[{Cos[π / 4], Sin[π / 4]}], PointSize[Large], Red], Axes -> True, Frame -> False, AxesLabel -> {x, y}] Out[1]= [image] ``` ### Properties & Relations (13) Basic parity and periodicity properties are automatically applied: ```wl In[1]:= Sin[x + 2Pi] Out[1]= Sin[x] In[2]:= Sin[-x] Out[2]= -Sin[x] In[3]:= Sin[I x] Out[3]= I Sinh[x] In[4]:= 1 / Sin[x] Out[4]= Csc[x] ``` --- Complicated expressions containing trigonometric functions do not simplify automatically: ```wl In[1]:= Sin[3z]^2 + (2 Cos[z] Cos[2 z] - Cos[z])^2 Out[1]= (-Cos[z] + 2 Cos[z] Cos[2 z])^2 + Sin[3 z]^2 In[2]:= Simplify[%] Out[2]= 1 In[3]:= Sin[x] - Sin[y] - 2Cos[(x + y/2)]Sin[(x - y/2)] Out[3]= Sin[x] - 2 Cos[(x + y/2)] Sin[(x - y/2)] - Sin[y] In[4]:= Simplify[%] Out[4]= 0 ``` --- Compose with inverse functions: ```wl In[1]:= {Sin[ArcSin[z]], Sin[2ArcSin[z]], Sin[3ArcSin[z]]} Out[1]= {z, Sin[2 ArcSin[z]], Sin[3 ArcSin[z]]} In[2]:= FunctionExpand[%] Out[2]= {z, 2 Sqrt[1 - z] z Sqrt[1 + z], 3 z - 4 z^3} ``` --- 1 radian is $\frac{180}{\pi }$ degrees: ```wl In[1]:= Sin[1] == SinDegrees[180 / π]//Simplify Out[1]= True ``` --- Solve a trigonometric equation: ```wl In[1]:= Reduce[Sin[z]^2 + 3 Sin[z + Pi / 6] == 4, z] Out[1]= C[1]∈ℤ && (z == 2 ArcTan[Root[{-3 + #1^2 & , 5 - (6*#1)*#2 + 8*#2^2 - (6*#1)*#2^3 + 11*#2^4 & }, {2, 1}]] + 2 π C[1] || z == 2 ArcTan[Root[{-3 + #1^2 & , 5 - (6*#1)*#2 + 8*#2^2 - (6*#1)*#2^3 + 11*#2^4 & }, {2, 2}]] + 2 π C[1] || z == 2 ArcTan[Root[{-3 + #1^2 & , 5 - (6*#1)*#2 + 8*#2^2 - (6*#1)*#2^3 + 11*#2^4 & }, {2, 3}]] + 2 π C[1] || z == 2 ArcTan[Root[{-3 + #1^2 & , 5 - (6*#1)*#2 + 8*#2^2 - (6*#1)*#2^3 + 11*#2^4 & }, {2, 4}]] + 2 π C[1]) ``` --- Numerically find a root of a transcendental equation: ```wl In[1]:= FindRoot[Sin[z]^2 + 3 Sin[z + Pi / 6] == z, {z, 2}] Out[1]= {z -> 2.12893} ``` --- Reduce a trigonometric equation: ```wl In[1]:= Reduce[Sin[α x + β] == 0, x] Out[1]= C[1]∈ℤ && ((α == 0 && (β == 2 π C[1] || β == π + 2 π C[1])) || (α ≠ 0 && (x == (-β + 2 π C[1]/α) || x == (π - β + 2 π C[1]/α)))) ``` --- Fourier transform: ```wl In[1]:= FourierTransform[Sin[t], t, s] Out[1]= I Sqrt[(π/2)] DiracDelta[-1 + s] - I Sqrt[(π/2)] DiracDelta[1 + s] In[2]:= LaplaceTransform[Sin[t], t, s] Out[2]= (1/1 + s^2) ``` --- ``Sin`` appears in special cases of many mathematical functions: ```wl In[1]:= {BesselJ[(1/2), z], MathieuS[1, 0, z], JacobiSN[z, 0], HypergeometricPFQ[{}, {(3/2)}, -z], MeijerG[{{}, {}}, {{(1/2)}, {0}}, z]} Out[1]= {(Sqrt[(2/π)] Sin[z]/Sqrt[z]), Sin[z], Sin[z], (Sin[2 Sqrt[z]]/2 Sqrt[z]), (Sin[2 Sqrt[z]]/Sqrt[π])} ``` --- ``Sin`` is a numeric function: ```wl In[1]:= NumericQ[Sin[2 + E]] Out[1]= True ``` --- ``Sin`` can be represented as a ``DifferentialRoot`` : ```wl In[1]:= DifferentialRootReduce[Sin[x], x] Out[1]= DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {\[FormalY][\[FormalX]] + Derivative[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][0] == 0, Derivative[1][\[FormalY]][0] == 1}]][x] ``` --- The generating function for ``Sin`` : ```wl In[1]:= GeneratingFunction[Sin[n], n, x] Out[1]= (x Sin[1]/1 + x^2 - 2 x Cos[1]) In[2]:= Series[%, {x, 0, 5}]//FullSimplify Out[2]= SeriesData[x, 0, {Sin[1], Sin[2], Sin[3], Sin[4], Sin[5]}, 1, 6, 1] ``` --- The exponential generating function for ``Sin`` : ```wl In[1]:= ExponentialGeneratingFunction[Sin[n], n, x] Out[1]= Sin[x Sin[1]] (Cosh[x Cos[1]] + Sinh[x Cos[1]]) ``` ### Possible Issues (6) Machine-precision input is insufficient to get a correct answer: ```wl In[1]:= Sin[10. ^ 30] Out[1]= 0.00933147 ``` With exact input, the answer is correct: ```wl In[2]:= N[Sin[10 ^ 30], 20] Out[2]= -0.090116901912138058030 ``` --- A larger setting for ``\$MaxExtraPrecision`` can be needed: ```wl In[1]:= N[Sin[10 ^ 100], 20] ``` N::meprec: Internal precision limit \$MaxExtraPrecision = 50.\` reached while evaluating Sin[10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]. ```wl Out[1]= 0``0 In[2]:= Block[{$MaxExtraPrecision = 200}, N[Sin[10 ^ 100], 20]] Out[2]= -0.37237612366127668826 ``` --- Machine‐number inputs can give high‐precision results: ```wl In[1]:= Sin[10. ^ 3I] Out[1]= 0.  + 9.8503555700852349694443967612`15.954589770191005*^433 I In[2]:= MachineNumberQ[%] Out[2]= False ``` --- Use ``FunctionExpand`` to express sine of rationals times $\pi$ using radicals: ```wl In[1]:= {Sin[Pi / 8], Sin[Pi / 12], Sin[Pi / 15]} Out[1]= {Sin[(π/8)], (-1 + Sqrt[3]/2 Sqrt[2]), Sin[(π/15)]} In[2]:= FunctionExpand[%] Out[2]= {(Sqrt[2 - Sqrt[2]]/2), (-1 + Sqrt[3]/2 Sqrt[2]), -(1/8) Sqrt[3] (-1 + Sqrt[5]) + (1/4) Sqrt[(1/2) (5 + Sqrt[5])]} ``` --- Continuous functions involving ``Sin[x]`` can give discontinuous indefinite integrals: ```wl In[1]:= Integrate[(1/2 + Sin[x]), x] Out[1]= (2 ArcTan[(1 + 2 Tan[(x/2)]/Sqrt[3])]/Sqrt[3]) In[2]:= Plot[%, {x, 0, 2Pi}] Out[2]= [image] ``` --- In ``TraditionalForm``, parentheses are needed around the argument: ```wl In[1]:= $$\sin x$$ Out[1]= sin x In[2]:= $$\sin (x)$$ Out[2]= Sin[x] ``` ### Neat Examples (5) Noncommensurate waves (quasiperiodic function): ```wl In[1]:= Plot[Sin[x] + Sin[Sqrt[2]x], {x, 0, 40Pi}] Out[1]= [image] ``` --- Some arguments can be expressed as a finite sequence of nested radicals: ```wl In[1]:= Sin[(π/2^12)]//FunctionExpand Out[1]= (1/2) Sqrt[2 - Sqrt[2 + Sqrt[2 + Sqrt[2 + Sqrt[2 + Sqrt[2 + Sqrt[2 + Sqrt[2 + Sqrt[2 + Sqrt[2 + Sqrt[2]]]]]]]]]]] ``` --- Indefinite integral of $\sin \left(x^n\right)$ : ```wl In[1]:= Integrate[Sin[x ^ n], x] Out[1]= -(I x (-(-I x^n)^-1 / n Gamma[(1/n), -I x^n] + (I x^n)^-1 / n Gamma[(1/n), I x^n])/2 n) ``` --- Chladni figure: ```wl In[1]:= DensityPlot[Sin[3x]Sin[5y] + 1 / 2Sin[5x]Sin[3y], {x, 0, 2Pi}, {y, 0, 2Pi}] Out[1]= [image] ``` --- Plot ``Sin`` at integer points: ```wl In[1]:= ArrayPlot[Table[Sin[x y], {x, -20, 20}, {y, -20, 20}]] Out[1]= [image] ``` ## See Also * [`AngleVector`](https://reference.wolfram.com/language/ref/AngleVector.en.md) * [`ArcSin`](https://reference.wolfram.com/language/ref/ArcSin.en.md) * [`Cos`](https://reference.wolfram.com/language/ref/Cos.en.md) * [`Tan`](https://reference.wolfram.com/language/ref/Tan.en.md) * [`Csc`](https://reference.wolfram.com/language/ref/Csc.en.md) * [`Degree`](https://reference.wolfram.com/language/ref/Degree.en.md) * [`SinDegrees`](https://reference.wolfram.com/language/ref/SinDegrees.en.md) * [`TrigToExp`](https://reference.wolfram.com/language/ref/TrigToExp.en.md) * [`TrigExpand`](https://reference.wolfram.com/language/ref/TrigExpand.en.md) * [`Sinc`](https://reference.wolfram.com/language/ref/Sinc.en.md) * [`Haversine`](https://reference.wolfram.com/language/ref/Haversine.en.md) * [`CirclePoints`](https://reference.wolfram.com/language/ref/CirclePoints.en.md) ## Tech Notes * [Some Mathematical Functions](https://reference.wolfram.com/language/tutorial/SomeMathematicalFunctions.en.md) * [Elementary Transcendental Functions](https://reference.wolfram.com/language/tutorial/MathematicalFunctions.en.md#10797) ## Related Guides * [Trigonometric Functions](https://reference.wolfram.com/language/guide/TrigonometricFunctions.en.md) * [GPU Computing](https://reference.wolfram.com/language/guide/GPUComputing.en.md) * [Precollege Education](https://reference.wolfram.com/language/guide/PrecollegeEducation.en.md) * [GPU Computing with Apple](https://reference.wolfram.com/language/guide/GPUComputing-Apple.en.md) * [GPU Computing with NVIDIA](https://reference.wolfram.com/language/guide/GPUComputing-NVIDIA.en.md) * [Functions for Separable Coordinate Systems](https://reference.wolfram.com/language/guide/FunctionsForSeparableCoordinateSystems.en.md) * [Mathematical Functions](https://reference.wolfram.com/language/guide/MathematicalFunctions.en.md) * [Elementary Functions](https://reference.wolfram.com/language/guide/ElementaryFunctions.en.md) ## Related Links * [MathWorld](http://mathworld.wolfram.com/Sine.html) * [An Elementary Introduction to the Wolfram Language: More about Numbers](https://www.wolfram.com/language/elementary-introduction/23-more-about-numbers.html) * [NKS\|Online](http://www.wolframscience.com/nks/search/?q=Sin) * [A New Kind of Science](http://www.wolframscience.com/nks/) ## History * Introduced in 1988 (1.0) \| Updated in 1999 (4.0) ▪ [2014 (10.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn100.en.md) ▪ [2015 (10.1)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn101.en.md) ▪ [2021 (13.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn130.en.md)