Age 14 to 18 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
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.