sec

Symbolic secant function

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Syntax

sec(X)

Description

sec(X) returns the secant function of X.

example

Examples

Secant Function for Numeric and Symbolic Arguments

Depending on its arguments, sec returns floating-point or exact symbolic results.

Compute the secant function for these numbers. Because these numbers are not symbolic objects, sec returns floating-point results.

A = sec([-2, -pi, pi/6, 5*pi/7, 11])

A =
   -2.4030   -1.0000    1.1547   -1.6039  225.9531

Compute the secant function for the numbers converted to symbolic objects. For many symbolic (exact) numbers, sec returns unresolved symbolic calls.

symA = sec(sym([-2, -pi, pi/6, 5*pi/7, 11]))

symA =
[ 1/cos(2), -1, (2*3^(1/2))/3, -1/cos((2*pi)/7), 1/cos(11)]

Use vpa to approximate symbolic results with floating-point numbers:

vpa(symA)

ans =
[ -2.4029979617223809897546004014201,...
-1.0,...
1.1547005383792515290182975610039,...
-1.6038754716096765049444092780298,...
225.95305931402493269037542703557]

Plot Secant Function

Plot the secant function on the interval from $- 4 π$ to $4 π$ .

syms x
fplot(sec(x),[-4*pi 4*pi])
grid on

Figure contains an axes object. The axes object contains an object of type functionline.

Handle Expressions Containing Secant Function

Many functions, such as diff, int, taylor, and rewrite, can handle expressions containing sec.

Find the first and second derivatives of the secant function:

syms x
diff(sec(x), x)
diff(sec(x), x, x)

ans =
sin(x)/cos(x)^2
 
ans =
1/cos(x) + (2*sin(x)^2)/cos(x)^3

Find the indefinite integral of the secant function:

int(sec(x), x)

ans =
log(1/cos(x)) + log(sin(x) + 1)

Find the Taylor series expansion of sec(x):

taylor(sec(x), x)

ans =
(5*x^4)/24 + x^2/2 + 1

Rewrite the secant function in terms of the exponential function:

rewrite(sec(x), 'exp')

ans =
1/(exp(-x*1i)/2 + exp(x*1i)/2)

Evaluate Units with `sec` Function

sec numerically evaluates these units automatically: radian, degree, arcmin, arcsec, and revolution.

Show this behavior by finding the secant of x degrees and 2 radians.

u = symunit;
syms x
f = [x*u.degree 2*u.radian];
secf = sec(f)

secf =
[ 1/cos((pi*x)/180), 1/cos(2)]

You can calculate secf by substituting for x using subs and then using double or vpa.

Input Arguments

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`X` — Input
symbolic number | symbolic scalar variable | symbolic matrix variable | symbolic expression | symbolic function | symbolic matrix function | symbolic vector | symbolic matrix

Input, specified as a symbolic number, scalar variable, matrix variable, expression, function, matrix function, or as a vector or matrix of symbolic numbers, scalar variables, expressions, or functions.

More About

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Secant Function

The secant of an angle, α, defined with reference to a right angled triangle is

$sec (α) = \frac{1}{\cos (α)} = \frac{hypotenuse}{adjacent side} = \frac{h}{b} .$

Right triangle with vertices A, B, and C. The vertex A has an angle α, and the vertex C has a right angle. The hypotenuse, or side AB, is labeled as h. The opposite side of α, or side BC, is labeled as a. The adjacent side of α, or side AC, is labeled as b. The secant of α is defined as the hypotenuse h divided by the adjacent side b.

The secant of a complex argument, α, is

$sec (α) = \frac{2}{e^{i α} + e^{- i α}} .$

Version History

Introduced before R2006a

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R2022a: Compute secant of symbolic matrix functions

The sec function accepts an input argument of type symfunmatrix.

R2021b: Compute secant of symbolic matrix variables

The sec function accepts an input argument of type symmatrix.

sec