Food for thought

## Things you might consider

Which is larger: $\sin(\cos x)$ or $\cos(\sin x)$?

Does it depend on $x$?

Let’s start by thinking about what we know about the behaviour of $\sin x$ and $\cos x.$

• Both $\sin x$ and $\cos x$ have period $2\pi$ so $\sin(\cos x)$ and $\cos(\sin x)$ will be periodic. The period of each function can’t be more than $2\pi$ so we only need to consider what happens for $x$ in the interval $[0,2\pi].$

• $\sin x$ and $\cos x$ vary between $-1$ and $1.$ What does this tell us about the minimum and maximum values of $\sin(\cos x)$ and $\cos(\sin x)$?

• For $x$ in $\big[0,\tfrac{\pi}{2}\big]$ and $\big[\tfrac{3\pi}{2},2\pi\big]$ $\sin x$ increases and for $x$ in $\big[\tfrac{\pi}{2},\tfrac{3\pi}{2}\big]$ it decreases. Where does $\cos x$ decrease or increase between $0$ and $2\pi$?

• We know that $\cos x$ is an even function and $\sin x$ is an odd function, i.e. $\cos (-x) = \cos x$ and $\sin(-x)=-\sin x.$

### Special points

We’ll consider some multiples of $\tfrac{\pi}{2}$ to get a sense of how the two functions behave. You may find it helpful to start marking some points on a sketch graph as you read.

• When $x=0, \pi, 2\pi$ the value of $\cos(\sin x)$ is $1$, which must be the maximum value of the function. This means the period of $\cos(\sin x)$ could be $\pi.$

• $\sin(\cos 2\pi)=\sin(\cos 0) = \sin 1.$ The value $\sin 1$ must be positive, but as $\sin x$ does not reach $1$ until $x=\tfrac{\pi}{2}$, $\sin(\cos 0) <1.$ Similarly $\sin (\cos \pi)=\sin (-1) = -\sin 1 >-1.$

• $\sin \tfrac{\pi}{2}=1$, so $\cos (\sin \tfrac{\pi}{2})=\cos 1$, which is positive, but less than $1.$ Also, $\cos (\sin \tfrac{3\pi}{2})=\cos 1$ because of the period of $\cos (\sin x).$

• $\cos x=0$ when $x=\tfrac{\pi}{2}$ and $x=\tfrac{3\pi}{2}$ so $\sin(\cos x)=0$ at these points.

This means that $\cos(\sin x) >\sin(\cos x)$ at $0,\tfrac{\pi}{2},\pi,\tfrac{3\pi}{2}$ and $2\pi.$ The values of $\cos 1$ and $\sin 1$ seem to be important. Without using a calculator or graphing software, how could you decide which of these values is bigger?

### Behaviour between special points

For all the points considered, $\cos(\sin x) >\sin(\cos x)$ and we now know a lot about the graphs $y=\cos (\sin x)$ and $y=\sin (\cos x).$ It seems reasonable to conjecture that $\cos(\sin x) >\sin(\cos x)$ for all $x$, but to convince ourselves that this is true, we need to show that the graphs do not cross anywhere.