gives the sine of z.
- 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 ; 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 can be used with Interval and CenteredInterval objects. »
- Sin automatically threads over lists.
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 be a radian angle measured counterclockwise from the 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 in a right triangle is the ratio of the length of the leg opposite to the length of the hypotenuse.
- Sin automatically evaluates to exact values when its argument is a simple rational multiple of . 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 , as reported by FunctionPeriod. Sin satisfies the identity , which is equivalent to the Pythagorean theorem. The definition of the sine function is extended to complex arguments using the definition , where 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 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.
Examplesopen allclose all
Basic Examples (5)
Use Degree to specify an argument in degrees:
Numerical Evaluation (6)
Sin can take complex number inputs:
Evaluate Sin efficiently at high precision:
Sin threads elementwise over lists and matrices:
Specific Values (6)
Values of Sin at fixed points:
Sin has exact values at rational multiples of pi:
More complicated cases require explicit use of FunctionExpand:
Zeros of Sin:
Extrema of Sin:
Plot the Sin function:
Function Properties (13)
Sin is defined for all real and complex values:
Sin achieves all real values between and 1:
Sin is a periodic function with a period :
Sin is an odd function:
Sin has the mirror property :
Sin is an analytic function of x:
Sin is monotonic in a specific range:
Sin is not injective:
Sin is not surjective:
Sin is neither non-negative nor non-positive:
Sin has no singularities or discontinuities:
Sin is neither convex nor concave:
Sin is concave for x in [0,π]:
Series Expansions (4)
Integral Transforms (3)
Function Identities and Simplifications (6)
Properties & Relations (12)
Sin appears in special cases of many mathematical functions:
Sin is a numeric function:
The generating function for Sin:
The exponential generating function for Sin:
Possible Issues (6)
A larger setting for $MaxExtraPrecision can be needed:
Use FunctionExpand to express sine of rationals times using radicals:
Continuous functions involving Sin[x] can give discontinuous indefinite integrals:
In TraditionalForm, parentheses are needed around the argument:
Neat Examples (5)
Plot Sin at integer points: