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Trigonometry, find the minimum of tan(a).tan(b).tan(c)

  1. Jun 19, 2011 #1
    1. The problem statement, all variables and given/known data

    1. Given a,b,c are acute angles and a + b + c =180. find the minimum of tan(a).tan(b).tan(c)

    2. Prove that if a+b+c=90, then tan(a)+tan(b)+tan(c) >= 31/2

    2. Relevant equations



    3. The attempt at a solution

    I don't even have any ideas how should i start to find/prove them... any hints?
     
    Last edited: Jun 19, 2011
  2. jcsd
  3. Jun 19, 2011 #2

    rock.freak667

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    For the first one I'd try to expand the left side after taking the tangent of both sides.

    tan(a+b+c) = tan(180°)
     
  4. Jun 20, 2011 #3

    ehild

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    Following rock.freak's suggestion, I would write the tangent of the third angle in terms of the tangents of the other two angles, and would use the relation between geometric and arithmetic means.

    ehild
     
  5. Jun 20, 2011 #4
    tan(a)*tan(b)*tan(c) = -tan(b+c) *tan(b)*tan(c) = (tanb+tanc)/(1-tanb*tanc) *tanb*tanc
    call S = tanb + tanc and P = tanb * tanc S^2>= 4P this should be easy from here on
     
  6. Jun 21, 2011 #5
    Still cannot do.... how bout question (2)? Any hints?
     
  7. Jun 21, 2011 #6
    you can use the langrange function to do it, it is quite easy if u use it
     
  8. Jun 21, 2011 #7

    ehild

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    We can not help if you do not show any attempt.

    ehild
     
  9. Jun 21, 2011 #8

    ehild

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    Note that it is Precalculus Math.

    ehild
     
  10. Jun 21, 2011 #9
    then just have to expand them to the sum and the product which is pretty much easier to solve
     
  11. Jun 21, 2011 #10

    rock.freak667

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    For the second one I would consider a right angled triangle and see if that can help.
     
  12. Jun 21, 2011 #11

    tiny-tim

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    Michael, are you trying?

    You need to minimise (tanb+tanc)/(1-tanb*tanc) *tanb*tanc …

    surely you have some idea how to do that?​
     
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