Extra (optional) exercises – Short Day

These exercises are optional extra material for students who have finished the exercises of the day and have the time and interest for more.

Extra exercise 1: Hyperbolic tangent

In Extra Exercise 2 from this week’s Long Day, the hyperbolic functions were introduced with the following expressions and both with domain and codomain \(\mathbb R\):

\[\mathrm{sinh}(x)=\frac{\mathrm e^x-\mathrm e^{-x}}{2}\quad\text{and}\mathrm{cosh}(x)=\frac{\mathrm e^x+\mathrm e^{-x}}{2}.\]

Question a

The symbols \(\mathrm{cosh}\) and \(\mathrm{sinh}\) were chosen because of the resemblance between these hyperbolic functions and the better-known trigonometric functions \(\cos\) and \(\sin\). A famous identity, sometimes known as the “dummy rule”, from the usual trigonometric functions is \(\cos^2(x)+\sin^2(x)=1\). Show that the corresponding, albeit different, identity for the hyperbolic functions is \(\mathrm{cosh}^2(x)-\mathrm{sinh}^2(x)=1\).

Note that while the point \((\cos(x),\sin(x))\) lies on the unit circle, the point \((\mathrm{cosh}(x),\mathrm{sinh}(x))\) lies on a hyperbola, hence the name.

Question b

Inspired by the formula \(\tan(x)=\sin(x)/\cos(x)\), we define \(\mathrm{tanh}(x)=\mathrm{sinh}(x)/\mathrm{cosh}(x)\) for all \(x \in \mathbb{R}\). The function \(\mathrm{tanh}: \mathbb{R} \to \mathbb{R}\) defined by \(x \mapsto \mathrm{tanh}(x)\) is called the hyperbolic tangent. What is the image set of this function?

Question c

Show that \(\mathrm{tanh}(x)=(\mathrm e^{2x}-1)/(\mathrm e^{2x}+1)\), and use this to find an expression for the inverse hyperbolic tangent function.

Hint

The function \(\mathrm{tanh}: \mathbb{R} \to \mathbb{R}\) is not surjective and thus has no inverse function. However, if we restrict the codomain to the image set \(B\) found in Question b, we obtain a function \(\mathrm{tanh}: \mathbb{R} \to B\) that does have an inverse. It is this inverse you are looking for.