# Wolfram Language & System 10.3 (2015)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# Cos

Cos[z]
gives the cosine of z.

## DetailsDetails

• Mathematical function, suitable for both symbolic and numerical manipulation.
• Unless explicitly given as a Quantity object, the argument of Cos is assumed to be in radians. (Multiply by Degree to convert from degrees.) »
• Cos 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, Cos automatically evaluates to exact values.
• Cos can be evaluated to arbitrary numerical precision.
• Cos automatically threads over lists.

## Background & ContextBackground & Context

• Cos is the cosine 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. Cos[x] then gives the horizontal coordinate of the arc endpoint. The equivalent schoolbook definition of the cosine of an angle in a right triangle is the ratio of the length of the leg adjacent to to the length of the hypotenuse.
• Cos 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. Cos[30 Degree]). When given exact numeric expressions as arguments, Cos may be evaluated to arbitrary numeric precision. Other operations useful for manipulation of symbolic expressions involving Cos include TrigToExp, TrigExpand, Simplify, and FullSimplify.
• Cos threads element-wise over lists and matrices. In contrast, MatrixFunction can be used to give the cosine of a square matrix (i.e. the power series for the cosine function with ordinary powers replaced by matrix powers) as opposed to the cosines of the individual matrix elements.
• Cos is periodic with period , as reported by FunctionPeriod. Cos satisfies the identity , which is equivalent to the Pythagorean theorem. The definition of the cosine function is extended to complex arguments using the definition , where is the base of the natural logarithm. The cosine function is entire, meaning it is complex differentiable at all finite points of the complex plane. Cos[z] has series expansion about the origin.
• The inverse function of Cos is ArcCos. The hyperbolic cosine is given by Cosh. Other related mathematical functions include Sec and Sin.

## ExamplesExamplesopen allclose all

### Basic Examples  (4)Basic Examples  (4)

The argument is given in radians:

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Use Degree to specify an argument in degrees:

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Plot over a subset of the reals:

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Series expansion:

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