Takes the hyperbolic sine of a number.

sinh(*value*)

**Menu Location**

- Press 2nd MATH to enter the MATH menu.
- Press C to enter the Hyperbolic submenu.
- Press 1 to select sinh(.

This command works on all calculators.

1 byte

The sinh() command returns the hyperbolic sine of a number.

As long as the calculator is in radian mode, sinh() can be used with complex numbers according to the rule that sinh(** i**x)=sin(x)

*****and sin(

*i***x)=sinh(x)**

*i******. This rule only works in radian mode, and sinh() of a complex number will return a domain error when working in degrees or gradians.

*i*Occasionally, sinh() can compute an exact result; most of the time, the calculator will leave an expression with sinh() alone unless it's in approximate mode (or you force an approximation). When sinh() is used with symbolic expressions, the calculator can go back and forth between the sinh() expression and its exponential equivalent.

```
:sinh(0)
0
:expand(sinh(x))
e^x/2-1/(2*e^x)
:comDenom(e^x/2-1/(2*e^x))
sinh(x)
```

If sinh() is applied to a list, it will take the hyperbolic sine of every element in the list.

# Advanced Uses

The sinh() of a matrix is not (in general) the same as taking the hyperbolic sine of every element of the matrix. A different definition is used to compute the result; see Matrices and Their Commands. It requires the matrix to be square and diagonalizable in order to apply.

# Formulas

The definition of hyperbolic sine is given in terms of exponents:

(1)# Error Conditions

**230 - Dimension** happens when taking sinh() of a matrix that isn't square.

**260 - Domain error** happens when taking sinh() of a complex number in degree or gradian mode.

**665 - Matrix not diagonalizable** happens when taking sinh() of a matrix that isn't diagonalizable.