csc

Symbolic cosecant function

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Syntax

csc(X)

Description

csc(X) returns the cosecant function of X.

example

Examples

Cosecant Function for Numeric and Symbolic Arguments

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

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

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

A =
   -1.0998   -1.0000    2.0000    1.2790   -1.0000

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

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

symA =
[ -1/sin(2), -1, 2, 1/sin((2*pi)/7), 1/sin(11)]

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

vpa(symA)

ans =
[ -1.0997501702946164667566973970263,...
-1.0,...
2.0,...
1.2790480076899326057478506072714,...
-1.0000097935452091313874644503551]

Plot Cosecant Function

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

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

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

Handle Expressions Containing Cosecant Function

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

Find the first and second derivatives of the cosecant function:

syms x
diff(csc(x), x)
diff(csc(x), x, x)

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

Find the indefinite integral of the cosecant function:

int(csc(x), x)

ans =
log(tan(x/2))

Find the Taylor series expansion of csc(x) around x = pi/2:

taylor(csc(x), x, pi/2)

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

Rewrite the cosecant function in terms of the exponential function:

rewrite(csc(x), 'exp')

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

Evaluate Units with `csc` Function

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

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

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

cosecf =
[ 1/sin((pi*x)/180), 1/sin(2)]

You can calculate cosecf 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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Cosecant Function

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

$csc (α) = \frac{1}{\sin (α)} = \frac{hypotenuse}{opposite side} = \frac{h}{a} .$

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 cosecant of α is defined as the hypotenuse h divided by the opposite side a.

The cosecant of a complex argument, α, is

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

Version History

Introduced before R2006a

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R2022a: Compute cosecant of symbolic matrix functions

The csc function accepts an input argument of type symfunmatrix.

R2021b: Compute cosecant of symbolic matrix variables

The csc function accepts an input argument of type symmatrix.

csc