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Sec
✖
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. »
  • Sec can be used with Interval and CenteredInterval objects. »

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 sum_(k=0)^infty((-1)^k TemplateBox[{{2, , k}}, EulerE])/((2 k)!)z^(2 k) 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

open allclose all

Basic Examples  (7)Summary of the most common use cases

The argument is given in radians:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-s2m
Out[1]=1

Use Degree to specify an argument in degrees:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-lx6
Out[1]=1

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-lxs
Out[1]=1

Plot over a subset of the reals:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-koz
Out[1]=1

Plot over a subset of the complexes:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-kiedlx
Out[1]=1

Series expansion at 0:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-unh
Out[1]=1

Asymptotic expansion at a singular point:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-klydni
Out[1]=1

Scope  (47)Survey of the scope of standard use cases

Numerical Evaluation  (5)

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-txv
Out[1]=1

The precision of the output tracks the precision of the input:

In[2]:=2
✖
https://wolfram.com/xid/0bud2i-xth5g
Out[2]=2

Evaluate for complex arguments:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-mw6
Out[1]=1

Evaluate Sec efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-di5gcr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-bq2c6r
Out[2]=2

Compute the elementwise values of an array using automatic threading:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-thgd2
Out[1]=1

Or compute the matrix Sec function using MatrixFunction:

In[2]:=2
✖
https://wolfram.com/xid/0bud2i-o5jpo
Out[2]=2

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-d49o1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-lmyeh7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bud2i-chi9ay
Out[3]=3

Or compute average-case statistical intervals using Around:

In[4]:=4
✖
https://wolfram.com/xid/0bud2i-cw18bq
Out[4]=4

Specific Values  (6)

Values of Sec at fixed points:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-nww7l
Out[1]=1

Values at infinity:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-bdij6w
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-drqkdo
Out[2]=2

Singular points of Sec:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-cw39qs
Out[1]=1

Local extrema of Sec:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-kb0ip3
Out[1]=1

Find a local minimum of Sec as a root of :

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-0hynlo
Out[1]=1

Substitute in the result:

In[2]:=2
✖
https://wolfram.com/xid/0bud2i-f2hrld
Out[2]=2

Visualize the result:

In[3]:=3
✖
https://wolfram.com/xid/0bud2i-khtscy
Out[3]=3

Simple exact values are generated automatically:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-du
Out[1]=1

More complicated cases require explicit use of FunctionExpand:

In[2]:=2
✖
https://wolfram.com/xid/0bud2i-izd
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bud2i-x8a
Out[3]=3

Visualization  (3)

Plot the Sec function:

In[2]:=2
✖
https://wolfram.com/xid/0bud2i-ecj8m7
Out[2]=2

Plot the real part of :

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-bo5grg
Out[1]=1

Plot the imaginary part of :

In[2]:=2
✖
https://wolfram.com/xid/0bud2i-d21315
Out[2]=2

Polar plot with :

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-epb4bn
Out[1]=1

Function Properties  (13)

The real domain of Sec:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-cl7ele
Out[1]=1

Complex domain:

In[2]:=2
✖
https://wolfram.com/xid/0bud2i-de3irc
Out[2]=2

Sec achieves all real values except the open interval :

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-evf2yr
Out[1]=1

Sec is a periodic function with a period :

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-ewxrep
Out[1]=1

Sec is an even function:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-dnla5q
Out[1]=1

Sec has the mirror property sec(TemplateBox[{z}, Conjugate])=TemplateBox[{{sec, (, z, )}}, Conjugate]:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-heoddu
Out[1]=1

Sec is not an analytic function:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-h5x4l2
Out[1]=1

However, it is meromorphic:

In[2]:=2
✖
https://wolfram.com/xid/0bud2i-e434t9
Out[2]=2

Sec is monotonic in a specific range:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-g6kynf
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-nlz7s
Out[2]=2

Sec is not injective:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-gi38d7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-ctca0g
Out[2]=2

Sec is not surjective:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-hkqec4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-hdm869
Out[2]=2

Sec is neither non-negative nor non-positive:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-84dui
Out[1]=1

It has both singularity and discontinuity when x is a multiple of π/2:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-mdtl3h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-mn5jws
Out[2]=2

Neither convex nor concave:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-kdss3
Out[1]=1

It is convex for x in [-1.5,1.5]:

In[2]:=2
✖
https://wolfram.com/xid/0bud2i-io426y
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bud2i-bb47uv
Out[3]=3

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-n2vtm

Differentiation  (3)

First derivative:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-mmas49
Out[1]=1

Higher derivatives:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-nfbe0l
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-fxwmfc
Out[2]=2

Formula for the ^(th) derivative:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-odmgl1
Out[1]=1

Integration  (3)

Indefinite integral of Sec:

In[2]:=2
✖
https://wolfram.com/xid/0bud2i-bponid
Out[2]=2

Definite integral of Sec over a period is 0:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-b9jw7l
Out[1]=1

More integrals:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-petr7o
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-fmyack
Out[2]=2

Series Expansions  (4)

Find the Taylor expansion using Series:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-ewr1h8
Out[1]=1

Plot the first three approximations for Sec around :

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-binhar
Out[2]=2

General term in the series expansion of Sec:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-dznx2j
Out[1]=1

Fourier series of Sec:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-f64drv
Out[1]=1

Sec can be applied to power series:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-f7dy9o
Out[1]=1

Function Identities and Simplifications  (6)

Sec of a double angle:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-mjplp7
Out[1]=1

Sec of a sum:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-nfe4y
Out[1]=1

Convert multiple‐angle expressions:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-jor
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-r64
Out[2]=2

Convert sums of trigonometric functions to products:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-s7f
Out[1]=1

Expand assuming real variables and :

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-pl4
Out[1]=1

Convert to complex exponentials:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-i6a
Out[1]=1

Function Representations  (4)

Representation through Sin:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-df304y
Out[1]=1

Representation through Bessel functions:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-ewa69v
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-9y9s4
Out[2]=2

Representation through SphericalHarmonicY:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-eeooyq
Out[1]=1

Representation in terms of MeijerG:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-6ksgj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-d3d3ld
Out[2]=2

Applications  (4)Sample problems that can be solved with this function

Generate a plot with poles removed:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-b4w
Out[1]=1

Generate a plot over the complex argument plane:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-d5v
Out[1]=1

Solve a differential equation:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-dz73fk
Out[1]=1

Pursuit curve in the reference frame of the predator with prey moving half as fast along a line:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-5ak5c
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-n4deyb
Out[2]=2

Properties & Relations  (11)Properties of the function, and connections to other functions

Basic parity and periodicity properties of the secant function get automatically applied:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-fpk
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-yh2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bud2i-ba4
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bud2i-jhh
Out[4]=4

Use TrigFactorList to factor Sec into Sin and Cos:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-iwb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-eul
Out[2]=2

Complicated expressions containing trigonometric functions do not autosimplify:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-wnm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-lxv
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bud2i-yow
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bud2i-qwt
Out[4]=4

Evaluate under additional assumptions:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-bhp
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-kd9
Out[2]=2

Compositions with the inverse functions:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-fk1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-s9l
Out[2]=2

1 radian is degrees:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-qtrjkp
Out[1]=1

Solve a trigonometric equation:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-sup
Out[1]=1

Solve for zeros and poles:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-6j
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-y7n
Out[2]=2

Numerically solve a transcendental equation:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-lj6
Out[1]=1

Sec is automatically returned as a special case for many mathematical functions:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-wag
Out[1]=1

Calculate residue symbolically and numerically:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-n85
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-tey
Out[2]=2

Possible Issues  (5)Common pitfalls and unexpected behavior

Machine-precision input is insufficient to give a correct answer:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-n9g
Out[1]=1

With exact input, the answer is correct:

In[2]:=2
✖
https://wolfram.com/xid/0bud2i-zi
Out[2]=2

A larger setting for $MaxExtraPrecision is needed:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-ldd
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-pxq
Out[2]=2

For arguments with imaginary part too large, the result cannot be represented by a computer:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-dcuu7v
Out[1]=1

The precision of the output can be much smaller or larger than the precision of the input:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-clnxgf
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-joyq7
Out[2]=2

In TraditionalForm, parentheses are needed around the argument:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-zcw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-uf
Out[2]=2

Neat Examples  (6)Surprising or curious use cases

Various integrals and products:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-waq
Out[1]=1
In[1]:=1
✖
https://wolfram.com/xid/0bud2i-i4t
Out[1]=1

Plot Sec at integer points:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-pwn
Out[1]=1
In[1]:=1
✖
https://wolfram.com/xid/0bud2i-tqw
Out[1]=1

Generate the Sec function from integrals and sums:

In[1]:=1
✖
https://wolfram.com/xid/0bud2i-cmv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bud2i-trv
Out[2]=2
In[1]:=1
✖
https://wolfram.com/xid/0bud2i-wco
Out[1]=1
Wolfram Research (1988), Sec, Wolfram Language function, https://reference.wolfram.com/language/ref/Sec.html (updated 2021).
✖
Wolfram Research (1988), Sec, Wolfram Language function, https://reference.wolfram.com/language/ref/Sec.html (updated 2021).

Text

Wolfram Research (1988), Sec, Wolfram Language function, https://reference.wolfram.com/language/ref/Sec.html (updated 2021).

✖
Wolfram Research (1988), Sec, Wolfram Language function, https://reference.wolfram.com/language/ref/Sec.html (updated 2021).

CMS

Wolfram Language. 1988. "Sec." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Sec.html.

✖
Wolfram Language. 1988. "Sec." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Sec.html.

APA

Wolfram Language. (1988). Sec. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Sec.html

✖
Wolfram Language. (1988). Sec. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Sec.html

BibTeX

@misc{reference.wolfram_2025_sec, author="Wolfram Research", title="{Sec}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Sec.html}", note=[Accessed: 01-May-2025 ]}

✖
@misc{reference.wolfram_2025_sec, author="Wolfram Research", title="{Sec}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Sec.html}", note=[Accessed: 01-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_sec, organization={Wolfram Research}, title={Sec}, year={2021}, url={https://reference.wolfram.com/language/ref/Sec.html}, note=[Accessed: 01-May-2025 ]}

✖
@online{reference.wolfram_2025_sec, organization={Wolfram Research}, title={Sec}, year={2021}, url={https://reference.wolfram.com/language/ref/Sec.html}, note=[Accessed: 01-May-2025 ]}