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.