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Homework Help: Differentiating trig

  1. Jan 12, 2010 #1
    1. The problem statement, all variables and given/known data


    we have been doing some error analysis in school, but they were very straight forward for example. centripetal acceleration : Fc = 4pi^2 m R / T^2


    however, for my project i must find the uncertainty in the angle that i measured.
    the angle is formed by a cardboard sitting on pile of books creating a triangle.
    Hypothenus = 33cm
    Height = Variable ( Changes according to the stack of the book)

    please help !! differentiate theta = inverseSin(Opposite/Hypothenus)






    2. Relevant equations



    theta = inverseSin(Opposite/Hypothenus)

    3. The attempt at a solution

    well since i measurd these distances with a ruler

    the uncertainty for hypothenus would be +- 0.005m (last half digit of a number)
    and the uncertainty for heght is the same + - 0.005m (last half dight of a number)

    how do i use these to differentiate for one another?
    which formurla woudl i use pleaes help !!
     
  2. jcsd
  3. Jan 12, 2010 #2

    Matterwave

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  4. Jan 12, 2010 #3
    i see that it is 1 ove squareroot of 1 minus x^2.
    but at my level of physics and calculus that is just different language to me.
    how can x be represented by (Opposite over hypothenus) ??
    how will i differentiate them? HELP !!!
     
  5. Jan 12, 2010 #4

    ideasrule

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    Homework Helper

    To do error analysis, you write the whole equation in terms of differentials; you don't differentiate with respect to anything. Here's an example:

    if 2x=y^2 and you differentiate both sides, you'd get 2dx=2ydy. Don't write it as dx/dy=y or dy/dx=1/y.

    For theta = inverseSin(Opposite/Hypothenus), you should get:

    d(theta)=1/sqrt(1-(o/h)^2) * derivative of o/h

    What do you get?
     
  6. Jan 12, 2010 #5
    hmm. 1*sqrt(1-(o/h)^-2)
    you would get negative exponent.

    is this correct so far? please guide me along the way.
    i would like to learn this by tom ~ help
     
  7. Jan 12, 2010 #6

    ideasrule

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    I don't get why it's 1*sqrt(1-(o/h)^-2). The derivative of inverse sine is 1/sqrt(1-x^2), so the first step in deriving theta = inverseSin(Opposite/Hypothenus) is 1/sqrt(1-(o/h)^2).

    After that, derive o/h. The answer would be (do*h + o*dh)/h^2, following the quotient rule. do and dh are the errors in the measured lengths and d(theta) is the error in the angle.
     
  8. Jan 13, 2010 #7
    so d(theta) = (do*h + o*dh)/h^2

    hmm.. do i sub in the numbers and that is the total uncertainty for theta?
    lets say i got 12 degrees from measuring sides with ruler and, after using the above equation i would write

    12 degrees + - (do*h + o*dh)/h^2
    would this be correct??

    but in my pratice sheet which is totally different than trig we have

    dF/dPi = 8PimR/T^2 from F = 4Pi^2mR/T^2

    and so on with all the other variables...

    but with this triangle is this it?
    or do i have to differentiate further.. please HELP !!
     
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