# cosh

Hyperbolic cosine

## Description

example

Y = cosh(X) returns the hyperbolic cosine of the elements of X. The cosh function operates element-wise on arrays. The function accepts both real and complex inputs. All angles are in radians.

## Examples

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Create a vector and calculate the hyperbolic cosine of each value.

X = [0 pi 2*pi 3*pi];
Y = cosh(X)
Y = 1×4
103 ×

0.0010    0.0116    0.2677    6.1958

Plot the hyperbolic cosine function over the domain $-5\le x\le 5.$

x = -5:0.01:5;
y = cosh(x);
plot(x,y)
grid on

The hyperbolic cosine satisfies the identity $\mathrm{cosh}\left(\mathit{x}\right)=\frac{{\mathit{e}}^{\mathit{x}}+{\mathit{e}}^{-\mathit{x}}}{2}$. In other words, $\mathrm{cosh}\left(\mathit{x}\right)$ is the average of ${\mathit{e}}^{\mathit{x}}$ and ${\mathit{e}}^{-\mathit{x}}$. Verify this by plotting the functions.

Create a vector of values between -3 and 3 with a step of 0.25. Calculate and plot the values of cosh(x), exp(x), and exp(-x). As expected, the curve for cosh(x) lies between the two exponential curves.

x = -3:0.25:3;
y1 = cosh(x);
y2 = exp(x);
y3 = exp(-x);
plot(x,y1,x,y2,x,y3)
grid on
legend('cosh(x)','exp(x)','exp(-x)','Location','bestoutside')

## Input Arguments

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Input angles in radians, specified as a scalar, vector, matrix, or multidimensional array.

Data Types: single | double
Complex Number Support: Yes

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### Hyperbolic Cosine

The hyperbolic cosine of an angle x can be expressed in terms of exponential functions as

$\mathrm{cosh}\left(x\right)=\frac{{e}^{x}+{e}^{-x}}{2}.$

In terms of the traditional cosine function with a complex argument, the identity is

$\mathrm{cosh}\left(x\right)=\mathrm{cos}\left(ix\right)\text{\hspace{0.17em}}.$

## Version History

Introduced before R2006a