Gosh cosh
Explore the hyperbolic functions sinh and cosh using what you know about the exponential function.
Problem
The hyperbolic trig functions $\cosh $ and $\sinh $ are defined by $$\eqalign { \cosh x &= {1\over 2}(e^x + e^{-x}) \cr \sinh x &= {1\over 2}(e^x - e^{-x}).}$$ Using the definitions sketch the graphs of $\cosh x$ and $\sinh x$ on one diagram and prove the hyperbolic trig identities $$\eqalign { \cosh^2 x - \sinh^2 x &=1 \cr \sinh 2x &= 2\sinh x \cosh x \cr \sinh (n+1)x &= \sinh nx \cosh x + \cosh nx \sinh x.}$$
Notice the strong resemblance of these formulae to standard trigonometrical identities. Using this similarity as a guide, investigate the properties of a 'hyperbolic tangent' function $tanh(x)$ defined by
$$\tanh(x)=\frac{\sinh(x)}{\cosh(x)}$$
NOTES AND BACKGROUND
Notice that the identities for hyperbolic functions that you have proved are very similar to the ordinary trigonometric identities. In fact there is a complete hyperbolic geometry with similar results to the trigonometric results in Euclidean geometry. We compare absolute values in the corresponding result for $\sin nx$ which is $|\sin nx|\leq n|\sin x|$ . This formula needs the absolute values because the function is periodic and takes negative values for some multiples of the angle. Notice that the inequality in $|\sin nx|\leq n|\sin x|$ goes the other way to the corresponding hyperbolic result. This is because $\cos x \leq 1$ for all $x$ whereas $\cosh x\geq 1$.
Getting Started
Teachers' Resources
Why do this problem?
This problem allows students to discover some of the properties of the hyperbolic functions for themselves rather than simply being presented with the results. Students do not need to have met hyperbolic trig functions before attempting this problem as the definitions are given and students are led through the question step by step. They will gain a greater understanding from thinking through these proofs for themselves.
Possible approach
This problem would follow on well from Hyperbolic thinking or even directly from Trig reps. Following on from one of these two problems might influence the way the definitions of $\cosh$ and $\sinh$ are introduced, but they can just be presented as functions to be explored. As such, this exploration is probably best conducted individually or perhaps in pairs. Further prompts can be found in the Getting Started section if needed.
When individuals have sketched the graphs and attempted to prove the given identities, their results can be compared and discussed, making a note of the most helpful strategies and observations as well as correcting any misconceptions.
Possible extension
Prove, by induction or otherwise, that for $x> 0$ and positive integral values of $n> 1$, $$\sinh nx > n\sinh x.$$