Hyperbolic cosine

Do you need calculate the hyperbolic cosine of a real number x? Use our calculator and you will automatically get the result of this hyperbolic function of which we will learn more in the following points.

To use the hyperbolic cosine calculator just type in the number x for which you want to get its cosh and press the calculate button to get the result.

Properties of the hyperbolic cosine

Graph of the hyperbolic cosine
Graph of the hyperbolic cosine

The hyperbolic cosine has a series of properties you should know in order to solve the hyperbolic functions exercises more easily.

Its domain ranges from (-∞, +∞) and it is a function. bijective in the codomain [1, +∞), even, convex and transcendent. All these properties can be seen reflected in the graphical representation of the function cosh (x) that you have a few lines above.

Hyperbolic Cosine

In addition, there are a number of relationships that are also very important:

  • cosh (x + y) = coshx coshy + senhx seny
  • cosh (x - y) = coshx coshy - senhx seny
  • cosh 2x = cosh2x + sinh2x
  • cosh2(x) = 1 + sinh2 (x)

Hyperbolic cosine of 0

Hyperbolic cosine formula

¿How much is the hyperbolic cosine of 0? To solve this question we are going to look at the formula of the function cosh(x) and that is the one we have just above these lines.

In this case we have to x = otherefore, we substitute and we have that:

cosh(0) = (e0 + e-0) / 2 = 1

This is so because any exponential raised to zero will give us 1 as a result, therefore, the hyperbolic cosine of 0 is equal to 1.

If you notice, the result that we have obtained fits perfectly with what we see in the graph of the function cosh (x)

Derivative of the hyperbolic cosine

Derivative of the hyperbolic cosine

The derivative of a function with cosh is equal to the formula you have just above these lines. If instead of a function it is a real number x, the derivative will be the following:

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

Therefore, the derivative of the hyperbolic cosine of x is equal to the hyperbolic sine of x.

Integral of the hyperbolic cosine

Integral of the hyperbolic cosine

At this point, it is also interesting for you to know about the integral of the hyperbolic cosine which, after the demonstration, will give us that it is equal to sinh (u) + c.

Calculate Cosh (x) in Excel

In addition to using our hyperbolic cosine calculator, you can also use our calculate the cosh of any real number using Excel and the following function:


In parentheses you have to write the number for which you want to calculate its cosh or point to a cell in which you have written that number.

Undoubtedly, it is another alternative method to check that the result of the exercise we have done is correct.

Do you have any doubts about the hyperbolic function cosh? If so, leave us a comment and we will try to help you as soon as possible.

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