# Sec

Sec[z]

gives the secant of z.

# Details

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

# Background & Context

• Sec is the secant function, which is one of the basic functions encountered in trigonometry. It is defined as the reciprocal of the cosine function: . It is defined for real numbers by letting be a radian angle measured counterclockwise from the axis along the circumference of the unit circle. Sec[x] then gives the reciprocal of the horizontal coordinate of the arc endpoint. The equivalent schoolbook definition of the secant of an angle in a right triangle is the ratio of the length of the hypotenuse to the length of the leg adjacent to .
• Sec 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 Sec 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. Sec[30 Degree]). When given exact numeric expressions as arguments, Sec may be evaluated to arbitrary numeric precision. Other operations useful for manipulation of symbolic expressions involving Sec include TrigToExp, TrigExpand, Simplify, and FullSimplify.
• Sec threads element-wise over lists and matrices. In contrast, MatrixFunction can be used to give the secant of a square matrix (i.e. the power series for the secant function with ordinary powers replaced by matrix powers) as opposed to the secants of the individual matrix elements.
• Sec is periodic with period , as reported by FunctionPeriod. Sec is periodic with period , as reported by FunctionPeriod. Sec satisfies the identity , which is equivalent to the Pythagorean theorem. The definition of the secant function is extended to complex arguments using the definition , where is the base of the natural logarithm. Sec has poles at for an integer and evaluates to ComplexInfinity at these points. Sec[z] has series expansion about the origin that may be expressed in terms of the Euler numbers EulerE.
• The inverse function of Sec is ArcSec. The hyperbolic secant is given by Sech. Other related mathematical functions include Cos and Csc.

# Examples

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## Basic Examples(5)

The argument is given in radians:

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Use Degree to specify an argument in degrees:

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Evaluate numerically:

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Plot over a subset of the reals:

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Series expansion:

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