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        riddles >> medium >> sin cos and cos sin
        
(Message started by: NickH on Jun 12th, 2004, 5:13am)
Title: sin cos and cos sin
Post by NickH on Jun 12th, 2004, 5:13am
Let x be a real number.  Which is greater: sin(cos(x)) or cos(sin(x))?
Title: Re: sin cos and cos sin
Post by BNC on Jun 12th, 2004, 5:56am
which goes to show that computer proofs have thei place: "solving" it graphically us very easy...
Title: Re: sin cos and cos sin
Post by Eigenray on Jun 12th, 2004, 10:28pm
First, we only need to consider x in [-[pi],[pi] by periodicity.
Second, [hide]because sin(cos(x)) and cos(sin(x)) are both even, we only need to consider x in [0, [pi].
If x is in [pi]/2, [pi], then sin(cos(x)) [le] sin(0) = 0 < cos(1) [le] cos(sin(x)).
So now suppose 0 < x < [pi]/2, and suppose sin(cos(x))=cos(sin(x)).
Let y = sin(x), z = cos(x), so that
0 < y,z < 1
sin(z) = cos(y) = sin(y+[pi]/2),
which means z = y+[pi]/2 +2n[pi], or z = [pi]-(y+[pi]/2) + 2n[pi], for some integer n.
In the first case,
-1 < z-y = [pi]/2 + 2n[pi] < 1,
which is impossible.  In the second case,
0 < y+z = [pi]/2 + 2n[pi] < 2,
which is possible only for n=0, which means:
[pi]/2 = y+z = sin(x)+cos(x) = [sqrt]2 sin(x+[pi]/4),
which is impossible because [pi]/2 > [sqrt]2.
And by the intermediate value theorem, if the function sin(cos(x))-cos(sin(x)) is never 0, it's either always positive or always negative.
So to answer the question, cos(sin(x))[/hide].
Title: Re: sin cos and cos sin
Post by srn347 on Sep 16th, 2007, 4:16pm
Also, cos(x)>sin(x). But what about sin2(x)+cos2(x) or cos4(x) + sin4(x) +2sin2(x)(cos2(x))?
Title: Re: sin cos and cos sin
Post by ThudanBlunder on Sep 16th, 2007, 5:13pm

on 09/16/07 at 16:16:00, srn347 wrote:
Also, cos(x)>sin(x). But what about sin2(x)+cos2(x) ?

cos(pi/3) < sin(pi/3)
sin2x + cos2x = 1
Title: Re: sin cos and cos sin
Post by srn347 on Sep 16th, 2007, 5:51pm
Check again. My calculator says otherwise about the pi/3 thing. That sin2(x)+cos2(x)=1 thing is true and in the question I asked, it is useful in finding the answer.
Title: Re: sin cos and cos sin
Post by ThudanBlunder on Sep 16th, 2007, 6:01pm

on 09/16/07 at 17:51:24, srn347 wrote:
Check again. My calculator says otherwise about the pi/3 thing.

Change your setting from Degrees to Radians and we'll be on the same page, kid.
Title: Re: sin cos and cos sin
Post by srn347 on Sep 16th, 2007, 6:29pm
It isn't that kind of calculator, but ok. How is that though? Shouldn't the series be the same? What about the other question I asked(which is higher:sin2(x)+cos2(x) or sin4(x)+cos4(x)+2sin2(x)cos2(x))?
Title: Re: sin cos and cos sin
Post by Aryabhatta on Sep 16th, 2007, 9:04pm

on 09/16/07 at 18:29:04, srn347 wrote:
What about the other question I asked(which is higher:sin2(x)+cos2(x) or sin4(x)+cos4(x)+2sin2(x)cos2(x))?


Do you really think such a trivial question should be in medium?

One is the square of the other and both are 1.

Also, please don't use existing threads to post your questions, create new ones instead (and in the appropriate forum).
Title: Re: sin cos and cos sin
Post by srn347 on Sep 17th, 2007, 6:25am
Once the question on a thread has been answered correctly, what else is there to do other than ask another question. Your answer was correct though.
Title: Re: sin cos and cos sin
Post by Barukh on Sep 17th, 2007, 7:58am

on 09/17/07 at 06:25:58, srn347 wrote:
Once the question on a thread has been answered correctly, what else is there to do other than ask another question

Incorrect. Please start a different thread if you want to ask a different question. You may supply a link to a relevant existing thread, if you wish.
Title: Re: sin cos and cos sin
Post by towr on Sep 17th, 2007, 8:00am

on 09/17/07 at 06:25:58, srn347 wrote:
Once the question on a thread has been answered correctly, what else is there to do other than ask another question.
If the new question is relevant to the topic, that's fine. Otherwise, it's better to leave the thread alone and ask the question in its own (and properly named) thread, so people that are interested in finding it can find it.
Title: Re: sin cos and cos sin
Post by srn347 on Sep 17th, 2007, 5:59pm
The new question was relevant. What is radiens anyway?
Title: Re: sin cos and cos sin
Post by ThudanBlunder on Sep 17th, 2007, 7:23pm

on 09/17/07 at 17:59:54, srn347 wrote:
What is radiens anyway?

All you paid was the looking price. Lessons are extra. - Edward G. Robinson

One minute you are telling professional mathematicians, engineers, and computer scientists what's what, and the next minute you are asking what 'radiens' are.  

Title: Re: sin cos and cos sin
Post by Sameer on Sep 17th, 2007, 8:08pm
Apparently a trigonometry class is in order!! ::) Whatever happened to ignoring?
Title: Re: sin cos and cos sin
Post by Aryabhatta on Sep 17th, 2007, 8:17pm

on 09/17/07 at 20:08:02, Sameer wrote:
Whatever happened to ignoring?


I tried, but as there always seemed to be someone who responded, so I gave up.
Title: Re: sin cos and cos sin
Post by Aryabhatta on Sep 17th, 2007, 8:19pm

on 09/17/07 at 17:59:54, srn347 wrote:
The new question was relevant.


How so?
If your answer is "trignometry" and "sin and cos", it is a bunch of bullsh*t.

The question you asked was not relevant and trivial to boot. You just polluted this thread. Sorry for the harsh words, but that is the truth.
Title: Re: sin cos and cos sin
Post by ThudanBlunder on Sep 17th, 2007, 8:23pm

on 09/17/07 at 20:08:02, Sameer wrote:
Whatever happened to ignoring?

Ignore it is then.   :-X
Title: Re: sin cos and cos sin
Post by Aryabhatta on Sep 17th, 2007, 8:30pm

on 09/17/07 at 20:23:32, ThudanBlunder wrote:
Ignore it is then.   :-X


Sorry. I will start ignoring again.
Title: Re: sin cos and cos sin
Post by srn347 on Sep 17th, 2007, 8:40pm
Can any of you give me a straight answer?!

// resized fonts.
Oversized fonts will not make people pay more attention to you. It is simply annoying. --towr
Title: Re: sin cos and cos sin
Post by Sameer on Sep 17th, 2007, 8:45pm
Come to think of this topic, Where is Nick nowadays? He is missing like Icarus...
Title: Re: sin cos and cos sin
Post by srn347 on Sep 17th, 2007, 9:01pm
I wonder...
Title: Re: sin cos and cos sin
Post by ima1trkpny on Sep 17th, 2007, 9:15pm

on 09/17/07 at 20:45:47, Sameer wrote:
Come to think of this topic, Where is Nick nowadays? He is missing like Icarus...

Nick's last post was in Jan... he's been gone even longer than Icarus (maybe he just decided 4600 was a nice even number of posts? I did actually check cause I was worried and there was no sign of him in the Wichita obits... so I assume he is fine and possibly either just busy or as I said before really likes that number)
Raven too has been missing for awhile...
Title: Re: sin cos and cos sin
Post by mikedagr8 on Sep 18th, 2007, 12:30am

Quote:
Apparently a trigonometry class is in order!!

Hehe, just had a triple today.

A radian is a measurement around a circle. Now for someone to have known calculus, YOU NEED TO KNOW THIS BACK TO FRONT!!! Luckily for me, I have just completed the topic, so I am well aware of what is required, you on the other hand, need to learn a few things srn347. The faster the better.
Title: Re: sin cos and cos sin
Post by SMQ on Sep 18th, 2007, 6:10am

on 09/17/07 at 20:40:21, srn347 wrote:
Can any of you give me a straight answer?!

Sorry, we keep assuming you know how to use Wikipedia (http://en.wikipedia.org/wiki/Radian), Mathworld (http://mathworld.wolfram.com/Radian.html), or at least Google (http://www.google.com/search?q=radian) to answer simple questions for yourself.  A radian is a measure of angle equal to the arc length of a corresponding segment of a circle with unit radius.  Since the perimeter of a unit circle is 2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif, 360o = 2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif radians.

--SMQ
Title: Re: sin cos and cos sin
Post by ThudanBlunder on Sep 18th, 2007, 9:21am

on 09/17/07 at 21:15:53, ima1trkpny wrote:
Nick's last post was in Jan...

Who is 'Nick'?
I can find only one person with that name (and 4 posts).
Title: Re: sin cos and cos sin
Post by Sameer on Sep 18th, 2007, 9:32am

on 09/18/07 at 09:21:03, ThudanBlunder wrote:
Who is 'Nick'?
I can find only one person with that name (and 4 posts).


I meant NickH!!!
Title: Re: sin cos and cos sin
Post by ThudanBlunder on Sep 18th, 2007, 9:44am

on 09/18/07 at 09:32:17, Sameer wrote:
I meant NickH!!!

As he invites puzzles for his site, I duly emailed him an interesting one a while back.
But he never acknowledged it, let alone posted it.
And as his site was last updated on August 8th, I doubt if I will be sending a second.
Title: Re: sin cos and cos sin
Post by srn347 on Sep 19th, 2007, 6:01pm
Just like what I sent you, thudan(d)blunder.


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