# Thread: Which is larger?

1. ## Which is larger?

Let x be an element of R. Find which term has a larger value, sin(cos x) or cos(sin x)?

Any help or detailed working out would be appreciated!

2. Originally Posted by Mr Rayon
Let x be an element of R. Find which term has a larger value, sin(cos x) or cos(sin x)?

Any help or detailed working out would be appreciated!
Dear Mr Rayon,

Please refer the attached file. The blue line represents y=sin(cosx) while the green curve represents y=cos(sinx). From this graph it is clear that sin(cosx)<cos(sinx). But for the moment I dont have any idea of how prove this result.

3. Originally Posted by Mr Rayon
Let x be an element of R. Find which term has a larger value, sin(cos x) or cos(sin x)?

Any help or detailed working out would be appreciated!
Here is another view.

$\displaystyle sinx$ ranges from $\displaystyle -1\rightarrow\ 1$

$\displaystyle cos(sinx)$ therefore ranges from $\displaystyle cos(-1)=cos(1)=0.54\rightarrow\ cos(0)=1$

$\displaystyle cosx$ ranges from $\displaystyle -1\rightarrow\ 1$

$\displaystyle sin(cosx)$ therefore ranges from $\displaystyle sin(-1)=-0.84\rightarrow\ sin(1)=0.84$

A little more analysis will show they are never equal.

4. Originally Posted by Archie Meade
Here is another view.

$\displaystyle sinx$ ranges from $\displaystyle -1\rightarrow\ 1$

$\displaystyle cos(sinx)$ therefore ranges from $\displaystyle cos(-1)=cos(1)=0.54\rightarrow\ cos(0)=1$

$\displaystyle cosx$ ranges from $\displaystyle -1\rightarrow\ 1$

$\displaystyle sin(cosx)$ therefore ranges from $\displaystyle sin(-1)=-0.84\rightarrow\ sin(1)=0.84$

A little more analysis will show they are never equal.
Dear Archie Meade,

Just out of curiosity, with what software did you create your images???

5. Originally Posted by Sudharaka
Dear Archie Meade,

Just out of curiosity, with what software did you create your images???
Hi Sudharaka,

I only use the "pages" application in Apple iWorks!
in conjunction with a Mathtype program (very basic).

AM

6. Originally Posted by Mr Rayon
Let x be an element of R. Find which term has a larger value, sin(cos x) or cos(sin x)?

Any help or detailed working out would be appreciated!
Both functions are continuous and you can't find a value of x for which they are equal.
So, if one is bigger than the other for a particular value of x, it will be bigger for all values of x.

7. Here is a way we can develop the proof...

$\displaystyle cos(sinx)>0$

hence we need not be concerned with $\displaystyle sin(cosx)<0$

and the maximum value of $\displaystyle sin(cosx)$ is <1.

We only need consider both functions >0 which is in quadrant 1, where $\displaystyle 0<x<\frac{\pi}{2}$

In quadrant 1, $\displaystyle \displaystyle\ sin(cosx)=cos\left(\frac{\pi}{2}-cosx\right)$

Hence we are examining $\displaystyle \displaystyle\ cos(sinx)-cos\left(\frac{\pi}{2}-cosx\right)$

so we want to know if $\displaystyle sinx<\left[\frac{\pi}{2}-cosx\right]$

since in quadrant 1, $\displaystyle cos(angle)>cos(larger\;angle)$

$\displaystyle \frac{\pi}{2}>sinx+cosx$ ?

Hence we find the maximum value of $\displaystyle sinx+cosx$ by differentiating...

$\displaystyle cosx-sinx=0\Rightarrow\ cosx=sinx$ which happens at $\displaystyle x=\frac{\pi}{4}$ radians.

The maximum value of $\displaystyle sinx+cosx=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\f rac{2}{\sqrt{2}}=\sqrt{2}$

$\displaystyle \frac{\pi}{2}>\sqrt{2}\Rightarrow\ cos(sinx)>sin(cosx)$

8. Originally Posted by BobP
Both functions are continuous and you can't find a value of x for which they are equal.
So, if one is bigger than the other for a particular value of x, it will be bigger for all values of x.
Hi BobP,

Is this a theorem? Can you please tell me where you have encountered it?

9. Originally Posted by Sudharaka
Hi BobP,

Is this a theorem? Can you please tell me where you have encountered it?
If it isn't a theorem I'm happy to claim it !

Actually, I thought it was just common sense .

10. Hi everyone,

Below is a proof that I have formulated for the theorem BobP had given in post #6. If you find any mistake in it please do not hesitate to tell me. Using this result the above problem could be easily solved.

11. The graph of the two functions $\displaystyle sin(cosx)$ and $\displaystyle cos(sinx)$ or $\displaystyle cos(sinx)-sin(cosx)$ are a nice guide.

If we are presented with just the 2 functions, $\displaystyle cos(sinx)$ and $\displaystyle sin(cosx)$

and we want to compare them, how do we answer "what stands out ?"

12. Originally Posted by Archie Meade
The graph of the two functions $\displaystyle sin(cosx)$ and $\displaystyle cos(sinx)$ or $\displaystyle cos(sinx)-sin(cosx)$ are a nice guide.

If we are presented with just the 2 functions, $\displaystyle cos(sinx)$ and $\displaystyle sin(cosx)$

and we want to compare them, how do we answer "what stands out ?"
Dear Archie Meade,

I dont quite understand what you meant by saying ".......how do we answer what stands out?" Can you please explain this.

13. ## Which is larger?

Originally Posted by BobP
Both functions are continuous and you can't find a value of x for which they are equal.
So, if one is bigger than the other for a particular value of x, it will be bigger for all values of x.

This tread is getting old, but Mr. Ryan still seemed confused.

This may be a Lemma to BobP's theorem.

I hope you can agree that: If two functions, f(x) and g(x) are continuous, and if they are not equal for any x, then one of them is greater than the other for all x.

In this case let's see if we can solve:

$\displaystyle \sin(\cos(x))=\cos(\sin(x))$.

$\displaystyle \cos(\theta)=\sin(\theta+{\pi\over2})$,

so $\displaystyle \sin(\cos(x))=\sin(\sin(x)+{\pi\over2})$.

Then $\displaystyle \cos(x)=\sin(x)+{\pi\over2}$. (O.K., you could add integer multiple of $\displaystyle 2\pi$ to this, but that won't help.)

$\displaystyle \cos(x)-\sin(x)={\pi\over2}$

Use the identity: $\displaystyle \cos(x)-\sin(x)=\sqrt{2}\cos(x+{\pi\over4})$.

This gives us $\displaystyle \sqrt{2}\cos(x+{\pi\over4})={\pi\over2}$

The maximum value of the cosine is 1, and $\displaystyle \sqrt{2}<{\pi\over2}$, so $\displaystyle \sin(\cos(x))=\cos(\sin(x))$ has no solution.

Since $\displaystyle \sin(\cos(\pi))=\sin(-1)\approx -0.8415$ (It's definitely negative!)

and $\displaystyle \cos(\sin(\pi))=\cos(0)=1$,

it must be that $\displaystyle \sin(\cos(x))<\cos(\sin(x))$, for all x.

14. Maybe simplest and without reference to graphs is:

$\displaystyle cos(A+B)=cosAcosB-sinAsinB\Rightarrow\ cos\left(\frac{\pi}{2}-A\right)=sinA$

$\displaystyle cos(sinx)-sin(cosx)=cos(sinx)-cos\left(\frac{\pi}{2}-cosx\right)$

This is the difference of cosines.

$\displaystyle cosA-cosB=-2sin\left[\frac{A+B}{2}\right]sin\left[\frac{A-B}{2}\right]$

$\displaystyle \Rightarrow\ cos(sinx)-sin(cosx)=-2sin\left[\frac{sinx-cosx+\frac{\pi}{2}}{2}\right]sin\left[\frac{sinx+cosx-\frac{\pi}{2}}{2}\right]$

One could go further with trigonometry here.
An alternative is to examine the contents of the square brackets on the previous line.
We can examine the maximum and minimum values of these expressions.

$\displaystyle \frac{d}{dx}(sinx-cosx)=cosx+sinx=0\Rightarrow\ cosx=-sinx$

Referring to the unit-circle, this occurs when $\displaystyle x=\frac{3{\pi}}{4},\;\;x=\frac{7{\pi}}{4}$

(1) maximum value within the first square brackets

$\displaystyle x=\frac{3{\pi}}{4}\Rightarrow\frac{\frac{1}{\sqrt{ 2}}+\frac{1}{\sqrt{2}}+\frac{\pi}{2}}{2}=\frac{1}{ \sqrt{2}}+\frac{\pi}{4}$

$\displaystyle 0<\frac{1}{\sqrt{2}}+\frac{\pi}{4}<\frac{\pi}{2}\R ightarrow\ sin\left[\frac{sinx-cosx+\frac{\pi}{2}}{2}\right]>0$

(2) minimum value within the first square brackets

$\displaystyle x=\frac{7{\pi}}{4}\Rightarrow\frac{-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}+\frac{\pi}{2}}{2}=-\frac{1}{\sqrt{2}}+\frac{\pi}{4}$

$\displaystyle 0<-\frac{1}{\sqrt{2}}+\frac{\pi}{4}<\frac{\pi}{2}\Rig htarrow\ sin\left[\frac{sinx-cosx+\frac{\pi}{2}}{2}\right]>0$

$\displaystyle \frac{d}{dx}(sinx+cosx)=cosx-sinx=0\Rightarrow\ cosx=sinx\Rightarrow\ x=\frac{\pi}{4},\;\;x=\frac{5{\pi}}{4}$

(3) maximum value within the second square brackets

$\displaystyle x=\frac{\pi}{4}\Rightarrow\frac{\frac{1}{\sqrt{2}} +\frac{1}{\sqrt{2}}-\frac{\pi}{2}}{2}=\frac{1}{\sqrt{2}}-\frac{\pi}{4}$

$\displaystyle -\frac{\pi}{2}<\frac{1}{\sqrt{2}}-\frac{\pi}{4}<0\Rightarrow\ sin\left[\frac{sinx+cosx-\frac{\pi}{2}}{2}\right]<0$

(4) minimum value within the second square brackets

$\displaystyle x=\frac{5\pi}{4}\Rightarrow\frac{-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}-\frac{\pi}{2}}{2}=-\frac{1}{\sqrt{2}}-\frac{\pi}{4}$

$\displaystyle -\frac{\pi}{2}<-\frac{1}{\sqrt{2}}-\frac{\pi}{4}<0\Rightarrow\ sin\left[\frac{cosx+sinx-\frac{\pi}{2}}{2}\right]<0$

Therefore, it follows that

$\displaystyle cos(sinx)-sin(cosx)>0\Rightarrow\ cos(sinx)>sin(cosx)$