csc

Symbolic cosecant function

Syntax

csc(X)

Description

csc(X) returns the cosecant function of X.

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

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

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

X — Input

symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix

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

More About

Cosecant Function

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

$$\text{csc}(\alpha )=\frac{1}{\sin \left(\alpha \right)}=\frac{\text{hypotenuse}}{\text{opposite side}}=\frac{h}{a}.$$

The cosecant of a complex argument, α, is

$$\text{csc}\left(\alpha \right)=\frac{2i}{e^{i\alpha }-e^{-i\alpha }}.$$