Expressing hyperbolic functions in terms of $e$.

Question

Express $\tanh(-3)$ in terms of $e$, where $\tanh$ is the hyperbolic tangent.

This is what I did:

$$\begin{align} \tanh(-x)&=\dfrac{e^{-2x}-1}{e^{-2x}+1}\\\\\\ \tanh(-3)&=\dfrac{e^{-2\times-3}-1}{e^{-2\times-3}+1}\\\\\\ \tanh(-3)&=\dfrac{e^6-1}{e^6+1} \end{align}$$

However, this is wrong, as the actual solution is:

$$\tanh(-3)=-\dfrac{e^3-1}{e^3+1}$$

1. What have I done that is unacceptable, hence making my solution wrong?

2. How is the actual solution obtained? (Full explanation would be helpful)

Answer 1

Using the definition $$\tanh(x) = \frac{e^{2x}-1}{e^{2x}+1}$$So we plug in $-3$ wherever we see an $x$ to get that $$\tanh(-3) = \frac{e^{2 \cdot-3}-1}{e^{2\cdot-3}+1}=\frac{e^{-6}-1}{e^{-6}+1} $$So we multiply by $\frac{e^6}{e^6}$ to get $$= \frac{1-e^6}{1 + e^6}$$So other than a little minus sign error, I think you're correct!

Answer 2

Your first attept is really right. In fact, you got $$\tanh(-3)=-\frac{e^3-e^{-3}}{e^3+e^{-3}}=-\frac{e^3-\frac{1}{e^{3}}}{e^3+\frac{1}{e^{-3}}}=-\frac{e^6-1}{e^6+1}$$ Knowing that $\tanh(x)$is an odd function also, the actual solution you pointed doesn't seem right result.