sinh function

Martin McBride
2020-09-10

The sinh function is a hyperbolic function. It is also known as the hyperbolic sine function.

Here is a video that explains sinh, cosh and tanh:

Equation and graph

The sinh function is defined as:

$$ \sinh{x} = \frac{e^{x}-e^{-x}}{2} $$

Here is a graph of the function:

sinh as average of two exponentials

The sinh function can be interpreted as the average of two functions, $e^{x}$ and $-e^{-x}$. This animation illustrates this:

Other forms of the equation

If we multiply the top and bottom of the original equation for the sinh function by $e^{x}$ (see below) we get:

$$ \sinh{x} = \frac{e^{2x}-1}{2e^{x}} $$

Alternatively, if we multiply the top and bottom of the original equation for the sinh function by $e^{-x}$ we get:

$$ \sinh{x} = \frac{1-e^{-2x}}{2e^{-x}} $$

Derivation of other forms

To see how the two formulae above were derived, we start with the original definition of sinh:

$$ \sinh{x} = \frac{e^{x}-e^{-x}}{2} $$

Multiplying top and bottom by $e^{x}$ gives:

$$ \sinh{x} = \frac{e^{x}(e^{x}-e^{-x})}{2e^{x}} = \frac{e^{x}e^{x}-e^{x}e^{-x})}{2e^{x}} $$

Remember that $e^{x}e^{x}$ is $e^{2x}$. Also $e^{x}e^{-x}$ is 1. This gives:

$$ \sinh{x} = \frac{e^{2x}-1}{2e^{x}} $$

which is the second form of the sinh equation. The other alternative form is derived in a similar way.

See also

⇐ Previous Next ⇒

Join the GraphicMaths Newletter

Sign up using this form to receive an email when new content is added:

Popular tags

angle cartesian equation chord circle combinations cosh cosine cosine rule cube diagonal directrix ellipse equilateral triangle exterior angle focus horizontal hyperbola hyperbolic function interior angle inverse hyperbolic function isosceles triangle locus major axis minor axis normal parabola parametric equation permutations quadrilateral radius sine rule sinh sloping lines solving equations solving triangles square standard curves straight line graphs tangent tanh triangle vertical