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Age 16 to 18
Challenge Level

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:

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?