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.