Combining Errors for Trigonometric Functions: Understanding the Formula

[Unto]

This is just my own practise question.

I have a relationship of

r = \frac{D}{2}sin4\theta

The apparent error formula is now:

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


Using a standard combination of errors formula, I only get

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

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

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




Thank you for any help

 

[Unto]

K I've decided I should treat sin 4 /theta as a separate 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

 

[Mapes]

Does http://en.wikipedia.org/wiki/Propagation_of_uncertainty#Example" help?

 

[Unto]

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?

 

[Mapes]

Your function r is a nonlinear function of two variables, and the link shows how to determine the variance \sigma_r^2 (what you are calling (\Lambda r)^2). You don't need to expand the sine term, but you do need to know its derivative.

 

[Unto]

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?

 

[Mapes]

The key equation is

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

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

Your function is r(D,\theta). Try working through the entire equation.

 

[Unto]

Equation doesn't work, it's like I have to fluke my way to get the answer.