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Inequalities of cosine

  1. Oct 2, 2008 #1
    1. The problem statement, all variables and given/known data

    (cos(x))^p [tex]\leq[/tex] cos(px)

    0[tex]\leq[/tex]x[tex]\leq[/tex]pi half

    and p, 0[tex]\leq[/tex](not equal) p [tex]\leq[/tex](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 :confused:)

    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. jcsd
  3. Oct 3, 2008 #2
  4. Oct 3, 2008 #3

    tiny-tim

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    without LaTeX

    Hi Nanie! :smile:

    Or, without LaTeX:

    (cosθ)p ≤ cos(pθ)

    0 ≤ θ ≤ π/2

    0 < p < 1. :wink:
     
  5. Oct 3, 2008 #4
    :surprised:biggrin: jeje thanks

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

    First the professor...gave us (cos[tex]\theta[/tex])p[tex]\leq[/tex] cos(p[tex]\theta[/tex]) 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.
     
  6. Oct 3, 2008 #5

    HallsofIvy

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    Your English is excellent!

    Perhaps the simplest way to prove this is to use a Taylor's polynomial. [itex]cos(\theta)= 1- (1/2)\theta^2+ (1/4!)\theta^4-\cdot\cdot\cdot[/itex]
    so [itex]cos(\theta)\le 1- (1/2)\theta^2[/itex]. Now use the extended binomial theorem to take that to the p power. You only need the first two terms.
     
  7. Oct 3, 2008 #6
    i dont know how to used it, explain me please!
     
  8. Oct 3, 2008 #7
    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..
     
  9. Oct 3, 2008 #8

    tiny-tim

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    Hi medinap! Hi Nanie! :smile:

    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 …" :wink:)
     
  10. Oct 3, 2008 #9
    ok thanks.... I will try!!!!! jeje...I have one hour to think !
     
  11. Oct 3, 2008 #10
    F([tex]\theta[/tex]) = cos ([tex]\theta[/tex]) p- cos (p[tex]\theta[/tex])

    How I use this?

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

    F(theta) = cos (theta)p- cos (p(theta))
     
  12. Oct 3, 2008 #11

    tiny-tim

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    Hi Nanie! :smile:

    (what happened to that θ i gave you? :rolleyes:)

    What is F'(θ)? Is it positive or negative?
     
  13. Oct 9, 2008 #12
    Thanks!!!!!!

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



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


    We did it.....?????
     
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