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# cosh

Symbolic hyperbolic cosine function

## Description

example

cosh(X) returns the hyperbolic cosine function of X.

## Examples

### Hyperbolic Cosine Function for Numeric and Symbolic Arguments

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

Compute the hyperbolic cosine function for these numbers. Because these numbers are not symbolic objects, cosh returns floating-point results.

A = cosh([-2, -pi*i, pi*i/6, 5*pi*i/7, 3*pi*i/2])
A =
3.7622   -1.0000    0.8660   -0.6235   -0.0000

Compute the hyperbolic cosine function for the numbers converted to symbolic objects. For many symbolic (exact) numbers, cosh returns unresolved symbolic calls.

symA = cosh(sym([-2, -pi*i, pi*i/6, 5*pi*i/7, 3*pi*i/2]))
symA =
[ cosh(2), -1, 3^(1/2)/2, -cosh((pi*2i)/7), 0]

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

vpa(symA)
ans =
[ 3.7621956910836314595622134777737,...
-1.0,...
0.86602540378443864676372317075294,...
-0.62348980185873353052500488400424,...
0]

### Plot Hyperbolic Cosine Function

Plot the hyperbolic cosine function on the interval from $-\pi$ to $\pi$.

syms x
fplot(cosh(x),[-pi pi])
grid on

### Handle Expressions Containing Hyperbolic Cosine Function

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

Find the first and second derivatives of the hyperbolic cosine function:

syms x
diff(cosh(x), x)
diff(cosh(x), x, x)
ans =
sinh(x)

ans =
cosh(x)

Find the indefinite integral of the hyperbolic cosine function:

int(cosh(x), x)
ans =
sinh(x)

Find the Taylor series expansion of cosh(x):

taylor(cosh(x), x)
ans =
x^4/24 + x^2/2 + 1

Rewrite the hyperbolic cosine function in terms of the exponential function:

rewrite(cosh(x), 'exp')
ans =
exp(-x)/2 + exp(x)/2

## Input Arguments

collapse all

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

## See Also

Introduced before R2006a

## Support

#### Mathematical Modeling with Symbolic Math Toolbox

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