# sinh

Symbolic hyperbolic sine function

## Description

example

sinh(X) returns the hyperbolic sine function of X.

## Examples

### Hyperbolic Sine Function for Numeric and Symbolic Arguments

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

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

A = sinh([-2, -pi*i, pi*i/6, 5*pi*i/7, 3*pi*i/2])
A =
-3.6269 + 0.0000i   0.0000 - 0.0000i   0.0000 + 0.5000i...
0.0000 + 0.7818i   0.0000 - 1.0000i

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

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

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

vpa(symA)
ans =
[ -3.6268604078470187676682139828013,...
0,...
0.5i,...
0.78183148246802980870844452667406i,...
-1.0i]

### Plot Hyperbolic Sine Function

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

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

### Handle Expressions Containing Hyperbolic Sine Function

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

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

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

ans =
sinh(x)

Find the indefinite integral of the hyperbolic sine function:

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

Find the Taylor series expansion of sinh(x):

taylor(sinh(x), x)
ans =
x^5/120 + x^3/6 + x

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

rewrite(sinh(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.