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Combination of Errors

  1. Jan 3, 2010 #1
    This is just my own practise question.

    I have a relationship of

    r = [tex]\frac{D}{2}[/tex]sin4[tex]\theta[/tex]

    The apparent error formula is now:

    [tex]\left(\frac{\Lambda r}{r}\right)^{2}[/tex] = [tex]\left(\frac{\Lambda D}{D}\right)^{2}[/tex] + 16 [tex]\left(\frac{\Lambda \theta}{tan 4 \theta}\right)^{2}[/tex]

    Using a standard combination of errors formula, I only get

    [tex]\left(\frac{\Lambda r}{r}\right)^{2}[/tex] = [tex]\left(\frac{\Lambda D}{D}\right)^{2}[/tex] + 16 [tex]\left(\frac{\Lambda \theta}{4 \theta}\right)^{2}[/tex]

    Since I know there would be an error in the measurement of the angle. The 16 I'm also unsure of anyway, I just guessed you had to square the 4 and put it on the outside. How to you do errors for trigonometric identities is what I want to know. How do you go from sin to tan and what is the reasoning behind what they did and came out with?

    Here is the error formula I think I am supposed to use

    [tex]\left(\frac{\Lambda F }{F}\right)^{2}[/tex] = [tex]k^{2} \left(\frac{\Lambda A}{A}\right)^{2}[/tex] + [tex] l^{2} \left(\frac{\Lambda B}{B}\right)^{2}[/tex]

    Thank you for any help
  2. jcsd
  3. Jan 3, 2010 #2
    K I've decided I should treat sin [tex]4 /theta[/tex] as a seperate piece. But it still doesn't help. What is Sine, how can I change it into a power?

    How do I solve this crap lol
  4. Jan 3, 2010 #3


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    Last edited by a moderator: Apr 24, 2017
  5. Jan 3, 2010 #4
    The symbols are confusing me sorry. Should I use a maclaurin series for sin (at least that way I would have a linear function).

    Can you please explain your link?
  6. Jan 3, 2010 #5


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    Your function r is a nonlinear function of two variables, and the link shows how to determine the variance [itex]\sigma_r^2[/itex] (what you are calling [itex](\Lambda r)^2[/itex]). You don't need to expand the sine term, but you do need to know its derivative.
  7. Jan 3, 2010 #6
    Lets do it step by step, the link has too much jargon and I simply can't understand it.

    K the derivative of sine 4 is 4 cos 4

    What importance does this have on our relationship?
  8. Jan 3, 2010 #7


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    The key equation is

    [tex]\sigma_f^2=\left(\frac{\partial f}{\partial a}\right)\sigma_a^2+\left(\frac{\partial f}{\partial b}\right)\sigma_b^2[/tex]

    for a function [itex]f(a,b)[/itex], where the errors are independent (i.e., the covariance [itex]\mathrm{COV}=0[/itex]).

    Your function is [itex]r(D,\theta)[/itex]. Try working through the entire equation.
  9. Jan 3, 2010 #8
    Equation doesn't work, it's like I have to fluke my way to get the answer.
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