Sunday, June 19, 2011

5.3 – Inverse Hyperbolic Functions

Inverse Hyperbolic Functions are obtained in the same way as the Inverse Trigonometric Functions. I think I don’t need to explain much, I’ll straight away show you the graphs:

cosh-1 x   sinh-1 x   tanh-1 x
              cosh-1 x                                            sinh-1 x                                      tanh-1 x

sech-1 x   csch-1 x   coth-1 x
                   sech-1 x                                       csch-1 x                                   coth-1 x

Note that due to the definition of functions, we only take the positive y values of the functions
cosh-1 x and sech-1 x. The domain and ranges are as follows:

There are not much formulas and identities for this section. But there is one very important thing that you are suppose to learn how to prove, which is the logarithmic form of inverse hyperbolic functions.

I’ll show you the proof for sinh-1 x:

Please promise me that you will learn how to prove the rest, this is super important.

Here are some identities to remember. Note that they are quite similar to the inverse trigonometric ones:


For all the above identities, please try to prove all of them. Refer to the section inverse trigonometric functions for some hints on the proofs.

That’s all for this chapter. Just remember how to proof them, sketch their graphs, and manipulate these functions. You will need to master this chapter before you can proceed to the next one…


  1. Can you prove the Inverse Hiperbolic for sech, please? I am having trouble. Please send it to my email. It would be much appreciated.

    Please. and Thank you.

    Email :

  2. I am also confused about how to do it for sech, please email me at

    Thank you in advance

  3. Thank you so much!
    I had hard time finding all these rules at one place!

  4. can you prove the inverse hyperbolic for sech and cosech.. please send it to my email, i really need ur help..
    please and tahnk you.