Hyperbolic thinking
Explore the properties of these two fascinating functions using trigonometry as a guide.
Age
16 to 18
Challenge level
Problem
This problem naturally follows on from Trig Reps, although the two problems may be attempted independently.
Steve left the following cryptic page in his notebook:
Image
It seems that Steve thinks the following functions $A(x)$ and $B(x)$ are similar in some way to $\sin(x)$ and $\cos(x)$:
$$A(x) = \frac{1}{2}\Big(10^{x} +10^{-x}\Big)\quad\quad B(x) = \frac{1}{2}\Big(10^{x} -10^{-x}\Big)$$
Is Steve correct? To answer this question, think of as many properties of $\sin(x)$ and $\cos(x)$ as you can and, using these as a guide, explore the properties of $A(x)$ and $B(x)$.
Once you have done this you might wish to consider the properties of functions similar to $A(x)$ and $B(x)$ where the $10$ is replaced by different numbers. Do any of the properties hold for all of the bases? Which properties are base dependent? Is there a natural choice of base which the structure reveals?
Getting Started
Start by looking at the values of $A^2(x), B^2(x), A(2x), B(2x), A(x)B(x)$ and see if you can spot any simple relationships linking them.
To differentiate the two functions use the expression
$$\frac{d}{dx}\left(a^x\right) = \ln(a) a^x$$
Consider simple values of the functions, such as $n\pi$ and $\pm 1$.
To differentiate the two functions use the expression
$$\frac{d}{dx}\left(a^x\right) = \ln(a) a^x$$
Consider simple values of the functions, such as $n\pi$ and $\pm 1$.
Student Solutions
Jin from Portugal sent in a solution investigating many properties of the new functions.
A^2(x) &= \frac{1}{4}\left(10^{2x} + 2 + 10^{-2x}\right) \\
B^2(x) &= \frac{1}{4}\left(10^{2x} - 2 + 10^{-2x}\right)
\end{align}
From which we deduce that $A^2(x) - B^2(x) = 1$ because subtracting in this way cancels the terms involving $x$, just leaving $1$. This analysis shows a similarity between these functions and the trigonometric ones, but whereas a point $\left(X, Y\right)$ with coordinates $\left(\cos x, \sin x\right)$ is on the unit circle with equation $X^2+Y^2 = 1$; a point $\left(A(x), B(x)\right)$ is on the unit hyperbola with equation $X^2-Y^2 = 1$.
\begin{align} A(x) &= A(-x) & B(x) = -B(-x) \end{align}
$$ 2A(x)B(x) = 2\left(\frac{1}{2}(10^x+10^-x)\frac{1}{2}(10^x-10^-x)\right) = \frac{1}{2}\left(10^{2x}-10^{-2x}\right)$$
which equals $B(2x)$ so the identity also holds with $A$ and $B$ swapped with $\cos$ and $\sin$ respectively.
$$ A^2(x) + B^2(x) = \frac{1}{2}\left(10^{2x} + 10^{-2x}\right) = A(2x)$$
So the identity for $A$ and $B$ follows the patters with a change of sign, just like in part 1. Very interesting.
\begin{align}
A(x+y) &= A(x)A(y) + B(x)B(y) \\
B(x+y) &= A(x)B(y) + B(x)A(y)
\end{align}
which are the $\cos$ formula with a sign change, and the $\sin$ formula respectively.
\begin{align}
\frac{d}{dx}A(x) &= \frac{d}{dx}\frac{1}{2}\left(10^x + 10^-x\right) \\
&= \frac{\ln {10}}{2}\left(10^x - 10^-x\right) \\
&= B(x)\ln {10}
\end{align}
You can work out $\frac{d}{dx}B(x)$ for yourself, try to spot the pattern.
\begin{align}
\frac{d}{dx}A(x) &= B(x) \ln b & \frac{d}{dx}B(x) &= A(x) \ln b
\end{align}
So it makes sense to choose $b$ such that $\ln b =1$. Then the relationships would all be like those for $\sin$ and $\cos$ except for a few signs swapped. The mathematical constant $e$ is the number which has $\ln e = 1$ and so we might consider the functions
\begin{align}
A(x) &= \frac{1}{2}\left(e^x+e^{-x}\right) & B(x) &= \frac{1}{2}\left(e^x-e^{-x}\right)
\end{align}
which are in fact called $\cosh x$ and $\sinh x$ (sometimes pronounced 'cosh' and 'shine'). These are the hyperbolic trigonometric functions.
1. A property similar to $\sin^2x + \cos^2x = 1$.
\begin{align}A^2(x) &= \frac{1}{4}\left(10^{2x} + 2 + 10^{-2x}\right) \\
B^2(x) &= \frac{1}{4}\left(10^{2x} - 2 + 10^{-2x}\right)
\end{align}
From which we deduce that $A^2(x) - B^2(x) = 1$ because subtracting in this way cancels the terms involving $x$, just leaving $1$. This analysis shows a similarity between these functions and the trigonometric ones, but whereas a point $\left(X, Y\right)$ with coordinates $\left(\cos x, \sin x\right)$ is on the unit circle with equation $X^2+Y^2 = 1$; a point $\left(A(x), B(x)\right)$ is on the unit hyperbola with equation $X^2-Y^2 = 1$.
2. Cosine is an even function, sine is odd.
Recall that $\cos(x) = \cos(-x)$ and $\sin(x) = -\sin(-x)$. This is also the case with $A$ and $B$ which have:\begin{align} A(x) &= A(-x) & B(x) = -B(-x) \end{align}
3. A property similar to $\sin(2x) = 2\sin x \cos x$.
To compare this identity first compute $2A(x)B(x)$$$ 2A(x)B(x) = 2\left(\frac{1}{2}(10^x+10^-x)\frac{1}{2}(10^x-10^-x)\right) = \frac{1}{2}\left(10^{2x}-10^{-2x}\right)$$
which equals $B(2x)$ so the identity also holds with $A$ and $B$ swapped with $\cos$ and $\sin$ respectively.
4. A property similar to $\cos(2x) = \cos^2 x - \sin^2 x$.
Look back at part 1. and see that we may add the expressions for $A^2(x)$ and $B^2(x)$ to cancel out the twos and leave$$ A^2(x) + B^2(x) = \frac{1}{2}\left(10^{2x} + 10^{-2x}\right) = A(2x)$$
So the identity for $A$ and $B$ follows the patters with a change of sign, just like in part 1. Very interesting.
5. Addition formulae like $\sin(x+y) = \sin x \cos y + \cos x \sin y$ and $\cos(x+y) = \cos x \cos y - \sin x \sin y$.
By now it feels like $A$ behaves like $\cos$ and $B$ like $\sin$ in these identities. Calculate the right hand sides of both versions, and correct the sign where necessary to find\begin{align}
A(x+y) &= A(x)A(y) + B(x)B(y) \\
B(x+y) &= A(x)B(y) + B(x)A(y)
\end{align}
which are the $\cos$ formula with a sign change, and the $\sin$ formula respectively.
6. Differentiation
We know that $\frac{d}{dx} \sin x = \cos x$ and $\frac{d}{dx} \cos x = -\sin x$ so do the same with $A$ and $B$. Recall that $\frac{d}{dx}a^x = a^x \ln a$. Then, for example,\begin{align}
\frac{d}{dx}A(x) &= \frac{d}{dx}\frac{1}{2}\left(10^x + 10^-x\right) \\
&= \frac{\ln {10}}{2}\left(10^x - 10^-x\right) \\
&= B(x)\ln {10}
\end{align}
You can work out $\frac{d}{dx}B(x)$ for yourself, try to spot the pattern.
7. The size of $\cos x$ and $\sin x$.
It is well known that $\left\vert \cos x \right\vert \leq 1$ and $\left\vert \sin x \right\vert \leq 1$ but what about $A$ and $B$? Well notice that for any real $x$ value the numbers $10^x$ and $10^{-x}$ are positive. Then what about very large $x$ values? We can make $10^x$ as big as we like by choosing big $x$ values. When $10^x$ is large; $10^{-x} = \frac{1}{10^x}$ is very small. This means that both $A(x)$ and $B(x)$ get as big as we want for large $x$ because the contribution from the $\pm 10^{-x}$ is tiny. This is rather unlike $\sin$ and $\cos$.8. Choice of base.
Most of the properties don't depend on the base but if we imagine replacing $10$ in the definitions by $b$, say, then we have:\begin{align}
\frac{d}{dx}A(x) &= B(x) \ln b & \frac{d}{dx}B(x) &= A(x) \ln b
\end{align}
So it makes sense to choose $b$ such that $\ln b =1$. Then the relationships would all be like those for $\sin$ and $\cos$ except for a few signs swapped. The mathematical constant $e$ is the number which has $\ln e = 1$ and so we might consider the functions
\begin{align}
A(x) &= \frac{1}{2}\left(e^x+e^{-x}\right) & B(x) &= \frac{1}{2}\left(e^x-e^{-x}\right)
\end{align}
which are in fact called $\cosh x$ and $\sinh x$ (sometimes pronounced 'cosh' and 'shine'). These are the hyperbolic trigonometric functions.
Teachers' Resources
Why do this problem
This problem introduces hyperbolic functions in an intriguing way which emphasises the natural links with trigonometric functions. It is a fairly well-signposted investigation which will provide practice in manipulation of powers, an understanding of the hyperbolic function identities and the importance of the base $e$ in the definitions. It also refreshes an area of the curriculum which can often feel dry or jaded - as the teacher, you might find some of the conclusions surprising or interesting (Steve certainly did, when he crafted the problem!).
Possible approach
You can use this problem as an introduction to hyperbolic functions or, with slight changes, as a consolidation of the topic. In both cases it will give a solid consolidation of the properties of the trigonometric functions. Here we give some suggestions for using the task to introduce hyperbolic functions.
Firstly project the page from Steve's notepad and give students a chance to read it. Open the discussion by asking 'Does anyone have any idea why Steve seems to think that these are like sine and cosine?'. Some students might note the similarities to the definition of sine and cosine in terms of exponentials, but this is not necessary prior knowledge. If nobody knows or spots the link, mention that it will be interesting to find out what the connection is! Even if the link is spotted, there are clearly significant differences between the two pairs of functions, which will make for an interesting analysis.
To address the main question, start by collectively listing as many of the properties of sine and cosine as possible - for your reference, the full list to aim for is that given in the question trig reps, but initially stick with those suggested by the students (Note that these properties fall into three rough categories: algebraic identities; differentiability properties; particular values.)
NOTE: It is not intended that students tackle this problem graphically. Pen, paper and analytical reasoning are the intended tools.
Once these have been collected encourage students to explore the properties of the functions $A(x)$ and $B(x)$, using the properties of the trig functions as a guide - give the class a nudge by suggesting that this might involve trying a few values, differentiating, squaring and adding, subtracting and so on.
You might want to draw up a table along these lines on the board which you get students to fill in individually or collectively as you see fit.
| Trigonometric property | Similar property of $A(x)$ and $B(x)$ | Similar or rather different? |
| $\cos(x)^2+\sin(x)^2 =1$ | ||
| $\cos(0) =1 ,\quad \sin(0)=0$ | ||
| $\cos(2x) = 2\cos^2(x)-1$ | ||
| $|\cos(x)|\leq 1, \quad |\sin(x)|\leq 1$ | ||
| $\frac{d}{dx}(\sin(x)) = \cos(x), \quad \frac{d}{dx}(\cos(x)) = -\sin(x)$ | ||
| etc ... |
Once the similarities and differences have been explored you might turn to the question of the base: in which areas is the base important? Is there a 'natural' choice of base? This sets the stage very nicely for the formal definitions of the hyperbolic functions.
Key questions
(use these only as prompts if the group is struggling to get started)
Compute and simplify $A^2(x), B^2(x), A(x)B(x), A(2x), B(2x)$. Can you spot any simple connections between these expressions?
What happens if you try to differentiate $A(x)$ and $B(x)$?
Possible extension
A fascinating related question is this: Is there an algebraic representation of sine and cosine if they are measured in degrees? Can you produce a 'family' of trigonometric-like functions? Alternatively, you might pose the question 'what if the $10$ is replaced by $i$?'.
Possible support
You can explicity ask "Can you work out $A^2(x)-B^2(x)?$" and "Can you relate $A(2x)$ to $A(x)$ and $B(x)$?. You might also need to show students how to differentiate $10^x$ if they have forgotten or don't yet know.