Is there a way to isolate x for tanh x = 1?

[silicon_hobo]

[SOLVED] is there a way to isolate x?

Homework Statement

Where does the tangent of tanh x = 1?

Homework Equations

$$\\f(x)=\tanh x$$
$$f^\prime(x)=\\sech^2x=1-\tanh^2x$$

The Attempt at a Solution

$$\\f^\prime(x)=1-\tanh^2x\rightarrow1-\tanh^2x=1\rightarrow\sqrt{\tanh^2x}=\sqrt{0}\rightarrow \tanh x=0$$

Since I have shown that the tangent = 1 when tanh x = 0, I thought it may be sufficient to simply add another line saying x = 0 since we know (based on the definition of tanh) that if tanh x = 0 then x = 0. However, I am wondering if there is a way to approach this to actually isolate x without using this assumption? Does that make sense? Thanks for reading.

 

[rocomath]

what does cosh and sinh equal in terms of exponentials?

 

[GregA]

You can...

as you said, $$\\sech^2x = 1 - \\tanh^2x$$
so for $$\\sech^2x = 1$$ then
$$1- \frac{e^x-e^{-x}}{e^x+e^{-x}} = 1$$

$$\Rightarrow$$ ...

 

[Oxymuon]

You know that tan(alpha) = 1 when alpha = pi/4. So now just write tanhx in terms of exponentials and solve using simple algebra (tanhx = pi/4).

 

[silicon_hobo]

Alright, I think I've got it. Continuing from where I left off:

$$0 = \tanh x =\frac{e^{2x}-1}{e^{2x}+1}$$

Therefore, $$e^{2x}-1=0$$ which gives $$x=0$$ as the only solution. Looks good yeah? Cheers.

 

[Oxymuon]

why are you setting tanhx = 0?...

 

[silicon_hobo]

So I could, in turn, use a trig identity to show that x = 0 when d/dx tanh x = 1. It should be correct if you read from the beginning. Would you happen to know how to mark posts as [SOLVED] do you? Cheers.