# sec

Symbolic secant function

## Description

example

sec(X) returns the secant function of X.

## 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 $-4\pi$ to $4\pi$.

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

### 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
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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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.

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### Secant Function

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

The secant of a complex argument, α, is

$\text{sec}\left(\alpha \right)=\frac{2}{{e}^{i\alpha }+{e}^{-i\alpha }}\text{\hspace{0.17em}}.$

## Version History

Introduced before R2006a

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