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!

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- November 28th 2010, 08:19 PMMr RayonWhich 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! - November 30th 2010, 06:29 AMSudharaka
- November 30th 2010, 08:04 AMArchie Meade
- November 30th 2010, 04:41 PMSudharaka
- November 30th 2010, 04:49 PMArchie Meade
- December 1st 2010, 01:42 AMBobP
- December 1st 2010, 05:39 AMArchie Meade
Here is a way we can develop the proof...

hence we need not be concerned with

and the maximum value of is <1.

We only need consider both functions >0 which is in quadrant 1, where

In quadrant 1,

Hence we are examining

so we want to know if

since in quadrant 1,

?

Hence we find the maximum value of by differentiating...

which happens at radians.

The maximum value of

- December 2nd 2010, 05:49 AMSudharaka
- December 4th 2010, 01:52 AMBobP
- December 4th 2010, 07:38 AMSudharaka
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. - December 4th 2010, 08:14 AMArchie Meade
The graph of the two functions and or are a nice guide.

If we are presented with just the 2 functions, and

and we want to compare them, how do we answer "what stands out ?" - December 4th 2010, 05:40 PMSudharaka
- December 4th 2010, 08:57 PMSammySWhich is larger?

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:

.

,

so .

Then . (O.K., you could add integer multiple of to this, but that won't help.)

Use the identity: .

This gives us

The maximum value of the cosine is 1, and , so has no solution.

Since (It's definitely negative!)

and ,

it must be that , for all x.

- December 31st 2010, 06:52 PMArchie Meade
Maybe simplest and without reference to graphs is:

This is the difference of cosines.

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.

Referring to the unit-circle, this occurs when

(1) maximum value within the first square brackets

(2) minimum value within the first square brackets

(3) maximum value within the second square brackets

(4) minimum value within the second square brackets

Therefore, it follows that