# Homework Help: Inequalities of cosine

1. Oct 2, 2008

### medinap

1. The problem statement, all variables and given/known data

(cos(x))^p $$\leq$$ cos(px)

0$$\leq$$x$$\leq$$pi half

and p, 0$$\leq$$(not equal) p $$\leq$$(not equal) 1

i need help, if some one can tell me how to started, what should i used i will really apreciate it!!!! (sorry for my english )

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Oct 3, 2008

### Nanie

http://img218.imageshack.us/img218/6451/ine111es8.jpg [Broken]

Last edited by a moderator: May 3, 2017
3. Oct 3, 2008

### tiny-tim

without LaTeX

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Hi Nanie!

Or, without LaTeX:

(cosθ)p ≤ cos(pθ)

0 ≤ θ ≤ π/2

0 < p < 1.

Last edited by a moderator: May 3, 2017
4. Oct 3, 2008

### Nanie

:surprised jeje thanks

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I'm friend of medinap....we really need help with this exercise!!!

First the professor...gave us (cos$$\theta$$)p$$\leq$$ cos(p$$\theta$$) but he don't gave us the values of p (we asume that n is for all natural p=2(2n+1)....but yesterday!!!!!!!!!!!!!!!!!!!! he made a correction that 0< p< 1

I don't know what to do

and sorry for my english too.

5. Oct 3, 2008

### HallsofIvy

Perhaps the simplest way to prove this is to use a Taylor's polynomial. $cos(\theta)= 1- (1/2)\theta^2+ (1/4!)\theta^4-\cdot\cdot\cdot$
so $cos(\theta)\le 1- (1/2)\theta^2$. Now use the extended binomial theorem to take that to the p power. You only need the first two terms.

6. Oct 3, 2008

### Nanie

i dont know how to used it, explain me please!

7. Oct 3, 2008

### Nanie

i know how to used the binomial theorem, but i dont know how to used it with taylor's polynomial i dont even know what that is..

8. Oct 3, 2008

### tiny-tim

Hi medinap! Hi Nanie!

Hint: differentiate (cosθ)p and cos(pθ), and remember that cos ≤ 1.

(btw, you both need to be more careful about using the past tense …

it's "if some one can tell me how to start, what should i use …"

and "he didn't gave us the values of p (we assumed that …" )

9. Oct 3, 2008

### Nanie

ok thanks.... I will try!!!!! jeje...I have one hour to think !

10. Oct 3, 2008

### Nanie

F($$\theta$$) = cos ($$\theta$$) p- cos (p$$\theta$$)

How I use this?

iah...I don't Know...

F(theta) = cos (theta)p- cos (p(theta))

11. Oct 3, 2008

### tiny-tim

Hi Nanie!

(what happened to that θ i gave you? )

What is F'(θ)? Is it positive or negative?

12. Oct 9, 2008

### Nanie

Thanks!!!!!!

F($$\theta$$) = p (sen$$\theta$$$$/$$cos$$^{1-p}$$) - sen(p$$\theta$$) > 0 ......however ...... cos (p$$\theta$$) > cos ($$\theta$$)$$^{p}$$

F'($$\theta$$) = p (sen$$\theta$$$$/$$cos$$^{1-p}$$ - sen p ($$\theta$$) > 0 (positive)

We did it.....?????