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

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

# Tan

Tan[z]
gives the tangent of .

## DetailsDetails

• Mathematical function, suitable for both symbolic and numerical manipulation.
• The argument of Tan is assumed to be in radians. (Multiply by Degree to convert from degrees.)
• Sin[z]/Cos[z] is automatically converted to Tan[z]. TrigFactorList[expr] does decomposition.
• Tan 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, Tan automatically evaluates to exact values.
• Tan can be evaluated to arbitrary numerical precision.
• Tan automatically threads over lists.

## Background & ContextBackground & Context

• Tan is the tangent function, which is one of the basic functions encountered in trigonometry. Tan[x] is defined as the ratio of the corresponding sine and cosine functions: . The equivalent schoolbook definition of the tangent of an angle in a right triangle is the ratio of the length of the leg opposite to the length of the leg adjacent to it.
• Tan 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. TrigFactorList can be used to factor expressions involving Tan into terms containing Sin and Cos. To specify an argument using an angle measured in degrees, the symbol Degree can be used as a multiplier (e.g. Tan[30 Degree]). When given exact numeric expressions as arguments, Tan may be evaluated to arbitrary numeric precision. Other operations useful for manipulation of symbolic expressions involving Tan include TrigToExp, TrigExpand, Simplify, and FullSimplify.
• Tan threads element-wise over lists and matrices. In contrast, MatrixFunction can be used to give the tangent of a square matrix (i.e. the power series for the tangent function with ordinary powers replaced by matrix powers) as opposed to the tangent of the individual matrix elements.
• Tan is periodic with period , as reported by FunctionPeriod. Tan satisfies the identity , which is equivalent to the Pythagorean theorem. The definition of the tangent function is extended to complex arguments using the definition , where is the base of the natural logarithm. Tan has poles at values for an integer and evaluates to ComplexInfinity at these points. Tan[z] has series expansion about the origin that may be expressed in terms of the Bernoulli numbers BernoulliB.
• The inverse function of Tan is ArcTan. The hyperbolic tangent is given by Tanh. Other related mathematical functions include Cot.

## 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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