# 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 ; 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.

# 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.

# Examples

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## Basic Examples(4)

The argument is given in radians:

 In:= Out= Use Degree to specify an argument in degrees:

 In:= Out= Plot over a subset of the reals:

 In:= Out= Series expansion:

 In:= Out= ## Neat Examples(5)

Introduced in 1988
(1.0)
|
Updated in 2015
(10.1)