Challenge Level

This is an investigation
concerning the composition of trig functions -- it is very
open-ended. You might wish to make good use of spreadsheets or
other tools to get started.

Two functions $f(x)$ and $g(x)$ can be composed to create a new function $h(x) = f(g(x))$.

Explore the properties of functions which can be created by composing two trig functions: $\sin(x)$, $\cos(x)$ and $\tan(x)$ on the range $-\pi < x \leq \pi$.

Which combinations are finite, which combinations have finite numbers of turning points and which combinations have no turning points on the specified range? What are their maximum and minimum values?

Extension: Explore the properties of nested sequences of $\sin$ and $\cos$: $\sin(\sin(\sin(x)))$ and $\cos(\cos(\cos(x)))$ or see what happens when you compose other functions.

Two functions $f(x)$ and $g(x)$ can be composed to create a new function $h(x) = f(g(x))$.

Explore the properties of functions which can be created by composing two trig functions: $\sin(x)$, $\cos(x)$ and $\tan(x)$ on the range $-\pi < x \leq \pi$.

Which combinations are finite, which combinations have finite numbers of turning points and which combinations have no turning points on the specified range? What are their maximum and minimum values?

Extension: Explore the properties of nested sequences of $\sin$ and $\cos$: $\sin(\sin(\sin(x)))$ and $\cos(\cos(\cos(x)))$ or see what happens when you compose other functions.