tanh^-1(x)=? Inverse hyperbolic tangent

Added: 2026-05-12

[00:00:00]Now, let's see how to come up with an expression for the inverse hyperbolic tangent. So, we will do the same thing that what we did last time for the inverse hyperbolic sine.

[00:00:09]We are going to start off by saying that this inverse hyperbolic tangent of X be, let's say, a variable T.

[00:00:18]Because this way we can apply the original hyperbolic tangent both sides.

[00:00:25]That way, this and that cancel, we can just get the X.

[00:00:29]Let me write this down first.

[00:00:32]Hyperbolic tangent of T is equal to just X.

[00:00:36]Once we have this, we can use the definition of the hyperbolic tangent, which is just hyperbolic sine over hyperbolic cosine, and both of them have like the denominator two, so they cancel.

[00:00:48]So, you will just get E to the T minus E to the negative T over E to the T plus E to the negative T, and that's equal to X.

[00:01:00]Now, we have T T T T four of them, so how do we solve it?

[00:01:05]We'll do the usual business. E to the negative T means one over E to the T.

[00:01:10]So, we get a little fraction inside of a big one, and this is also a fraction.

[00:01:15]Let's multiply the top and bottom by E to the T.

[00:01:21]So, this times this is E to the 2 T.

[00:01:24]This times this is just E to the zero's power, so we get one.

[00:01:29]Over the bottom is E to the 2 T, but plus one, and that's equal to X.

[00:01:37]Now, we have T here and T here, two places only, much better.

[00:01:41]To get T by itself, let's multiply this denominator both sides of the equation, so E to the 2 T plus one.

[00:01:50]So, when we take this times that, they cancel, we just get the top.

[00:01:56]And the right-hand side, when we do this times that, let me also distribute the X. So, we get X times E to the 2 T and then plus X.

[00:02:06]Now, remember we're trying to get T by itself.

[00:02:11]So, we will have to put all the terms with T on one side and then the rest on the other.

[00:02:16]Let me move this to there, so E to the 2 T, and then that would be a minus X E to the 2 T, and that's equal to bring the minus one to the other side, we get one plus X.

[00:02:32]Now, we can factor out E to the 2 T and we get one minus X.

[00:02:40]And then divide both sides by this, so we get E to the 2 T is equal to one plus X over one minus X.

[00:02:49]And then we'll just take the natural log both sides so that this and that cancel, and we get 2 T is equal to this. And usually we'll just keep it in the ln of this fraction form instead of breaking apart.

[00:03:05]And then lastly, we can just divide both sides by two, or multiply both sides by one half.

[00:03:11]So, we get T, which is equal to the inverse hyperbolic tangent of X, and that is equal to one half ln of one plus X over one minus X.

[00:03:25]Yep, just like this.

[00:03:28]And we're done.