# Making Waves

## Problem

- Sketch the graph of $y=\sin(\cos x)$, where $x$ is in radians.

Here are some things to consider when sketching your graph:It might help to start by sketching $y=\cos x$. What are the maximum and minimum values of $\cos x$?

Where does the graph cross the $y$ axis? (You don't have to find a numerical value!)

Can the graph cross the $x$ axis? If so, where?

Is the function even, odd or neither? A function is even if $f(-x)=f(x)$ and it is odd if $f(-x) = -f(x)$. An even function has reflection symmetry over the $y$ axis and an odd function has rotational symmetry about the origin.

The functions $\sin x$ and $\cos x$ are periodic with period $2 \pi$. What about $\sin(\cos x)$?

- Sketch the graph of $y=\cos(\sin x)$.

- Show that $\cos(\sin x) > \sin(\cos x)$ when $x=0, \pi$ and $\frac{\pi} 2$.

Can you also show that this is true when $x=\frac{\pi} 4$?

- Show that $\cos(\sin x) > \sin(\cos x)$ for all values of $x$.

*Some possible methods for this last part are discussed in the Getting Started Section.*

## Getting Started

It might be helpful to note that $1 < \frac {\pi} 2$.

From your sketches, you should be able to see that if you can prove the result for $x \in [0, \pi]$ then it will be true for all $x$. (You might like to compare your sketches with these graphs).

You should have already shown that $\cos(\sin x) > \sin(\cos x)$ when $x=0, \frac{\pi} 2$ and $\pi$.

Can you use your sketches to help you explain why $\cos(\sin x) > \sin(\cos x)$ when $x \in \left(\frac{\pi} 2, \pi \right)$?

The argument for $x\in \left( 0, \frac{\pi} 2\right)$ is a bit harder. One way to approach this is to try and prove the stronger result $\cos(\sin x) > \cos x > \sin(\cos x)$ for $x\in \left( 0, \frac{\pi} 2\right)$ instead.*You might like to sketch $y=\cos x$ alongside the other two graphs using Desmos to convince yourself this is true before trying to prove it!*

**Method 1: Calculus and increasing/decreasing functions**

Lemma: $x > \sin x$ for $x\in \left( 0, \frac{\pi} 2\right)$.

- Let $f(x)=x - \sin x$
- Can you show that $f(x)$ is an increasing function?
- If $f(x)$ is an increasing function, then explain why this means that $x < \sin x$ for $x\in \left( 0, \frac{\pi} 2\right)$.

We know that $\cos x$ is a decreasing function for $x\in \left( 0, \frac{\pi} 2\right)$. Explain why this, along with the lemma result, means that we have $\cos(\sin x) > \cos x$.

Explain why the lemma means that we have $\cos x > \sin(\cos x)$.

*A lemma is an intermediate theorem in the proof of another theorem.*

**Method 2: Geometry**

Imagine a quarter-circle where the radius is equal to $1$.

**
Image
**

- In Figure 1 explain why the arc length indicated is equal to $x$. Find the lengths of the dotted lines in terms of $x$. How does your diagram show that $x > \sin x$?
- In Figure 2 find $\alpha$ in terms of $x$, and also find the length of the extra dotted line. What inequality does this give you?

- Figure 3 is the same as Figure 1. Find the lengths of the dotted lines.
- In Figure 4 find angle $\beta$ in terms of $x$. Use this and the horizontal dotted lines to prove the second half of the inequality.

## Student Solutions

Well done to Dylan from Brooke Weston and Joshua from Bohunt Sixth Form in the UK who both sent full solutions.

Joshua sketched the graph of $y=\sin{(\cos x)}$ by considering the minimum and maximum values of $\cos x,$ where the function should be increasing and decreasing, and where the function should cross the co-ordinate axes:

Dylan sketched the graph of $y=\sin{(\cos x)}$ by considering the periodicity and symmetry of $\cos x$ as well as the minima, maxima and roots of the function:

Dylan and Joshua both used the same method they'd used before for the graph of $y=\cos{(\sin x)}.$ This is Joshua's work:

This is Dylan's work:

Dylan and Joshua both completed the rest of the problem - building up to showing that $\cos{(\sin x)}\gt\sin{(\cos x)}$ (or, in Dylan's words, $g(x)\gt f(x)$) for all $x.$ This is Dylan's work:

## Teachers' Resources

### Why do this problem?

This problem requires students to sketch graphs of trig functions, and think about the symmetries of these graphs. They are also required to think about inequalities and how they might show thee to be true.

For the last part of the problem there are a couple of suggested approaches in the Getting Started section.