# Hyperbolic sine calculator (sinh)

Input sinh and numbers in the above form to calculate the hyperbolic sine as follows.

The sinh is a command to calculate the hyperbolic sine and all numbers are separated by one space. The value that ends with d means a degree angle so 180d is about 3.141592... (pi).

• 90d ... 90°
• 180d ... 180°

The calculator takes radians, not degrees. So the degree numbers should have the suffix d and the calculator converts them to radians.

Let's input sinh and numbers in the form and calculate the hyperbolic sines. If you want to calculate hundreds or thousands of hyperbolic sines at once, use bulk calculator.

## The definition of hyperbolic sine

The following is the definition of hyperbolic sine.

$\textnormal{sinh} \ x = \dfrac{ e^x - e^{-x} }{2}$

If substituting x for -x in the function, you get -sinh().

\begin{aligned} \textnormal{sinh} (-x) &= \dfrac{ e^{-x} - e^{-(-x)} }{2} \\ &= \dfrac{ e^{-x} - e^x }{2} \\ &= - \textnormal{sinh} \ x \end{aligned}

The cosh() graph is symmetric with respect to y-axis but the sinh() graph is not.

## Hyperbolic sine formula

$\textnormal{sinh} (x+y) = \textnormal{sinh} \ x \ \textnormal{cosh} \ y + \textnormal{cosh} \ x \ \textnormal{sinh} \ y \\~\\ \textnormal{sinh} (x-y) = \textnormal{sinh} \ x \ \textnormal{cosh} \ y - \textnormal{cosh} \ x \ \textnormal{sinh} \ y$

$\textnormal{sinh} \ 2x = 2 \textnormal{sinh} \ x \textnormal{cosh} \ x$

$\textnormal{sinh} \left( \dfrac{x}{2} \right) = \dfrac{ \textnormal{sinh} \ x }{ \sqrt{ 2(\textnormal{cosh} \ x + 1) } }$

$\textnormal{sinh}^2 x = \dfrac{ \textnormal{cosh} \ 2x - 1 }{ 2 }$