Hyperbolic sinus

If you have to calculate the hyperbolic sine of a number real x, our online calculator provides you with the result of that calculation immediately.

Just enter the number x into the calculator and press the calculate button to obtain the hyperbolic sine. If you wish, we also have at your disposal another tool for calculating the hyperbolic cosine.

Hyperbolic sine formula

Hyperbolic sine formula

The hyperbolic sine of a real number x is computed by solving the formula above of these lines.

As you can see, it is important to be familiar with the equations exponentials.

Properties of the hyperbolic sine

Graph of the hyperbolic sine
Graph of the hyperbolic sine

The hyperbolic function sinh (x) has a series of properties you should know to be able to solve the exercises properly. Some of these properties are the following:

  • Its domain ranges from (-∞, +∞).
  • It is a bijective, increasing, transcendent and odd function.

In addition, it has these relations that will be very useful to you for the exercises:

  • sinh (x+y) = sinh coshy + coshx sinhy
  • sinh (x-y) = sinh coshy - coshx sinhy
  • sinh2x = 2senhx coshx
  • sinh2 (x) = cosh2(x) - 1

Hyperbolic sine of 0

Now that we know more about the hyperbolic function sinh (x), let's solve some typical exercises. In this case, we are going to calculate how much is sinh(0):

sinh(0) = (e0 - e-0) / 2 = (1 – 1) / 2  = 0

Therefore, we can verify that the hyperbolic sine of 0 equals zero.

Hyperbolic sine of 1

Now let's see what result we get by calculating sinh(1):

sinh(1) = (eπ - e) / 2 = (2,718 – 0,368) / 2 = 1,175

Hyperbolic sine of pi

Let's finish with these solved exercises of the hyperbolic sine calculating its value for π (pi):

sinh(π) = (e1 - e-1) / 2 = (23,139 – 0,043) / 2 = 11,548

If you have any problems withesolving the hyperbolic sine If you have any questions please leave us a comment and we will try to help you as soon as possible.

Derivative of the hyperbolic sine

Derivative of the hyperbolic sine

The derivative of senh (u) is equal to the hyperbolic cosine of that function multiplied by the derivative of its function. If instead of a function we have senh (x), where x is a real number, the derivative will be:

d/dx sinh(x) = cosh(x)

Integral of sinh (x)

Integral of the hyperbolic sine

In the same way, here we have the integral of senh (u).

Calculate hyperbolic sine in Excel

If you have the Excel spreadsheet program installed on your computer or mobile device, then you can also solve the hyperbolic cosine of a number using Microsoft software.

Just open a new spreadsheet and type in an empty cell the following formula:


Between the parentheses you must write the number for which you want to calculate the senh (x) or a cell in which that quantity is written. Either method is valid.

Inverse hyperbolic sine

Inverse hyperbolic sine

Finally, we leave you with the formula for the inverse hyperbolic sine as in some cases we may also be asked to do so.

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