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### Number and algebra

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### Advanced mathematics

# Trig-trig

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### Small Steps

Links to the University of Cambridge website
Links to the NRICH website Home page

Nurturing young mathematicians: teacher webinars

30 April (Primary), 1 May (Secondary)

30 April (Primary), 1 May (Secondary)

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Age 16 to 18

Challenge Level

- Problem

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.

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.