# Error propagation

1. Oct 24, 2009

### bigevil

1. The problem statement, all variables and given/known data

For my lab work, I have created a theoretical model that goes something like:

$$T = \sqrt{\frac{ks^2}{x \sin \theta \cos^2\theta}}$$

where k is a constant, and the variables to be differentiated are x, theta and s. How do I find the error of T? I can find the errors of x and s (dx/x and ds/s) from experiment. And of course

$$d\theta / \theta$$ can also be found from the instrument.

3. The attempt at a solution

Differentiating the term for theta,

$$\frac{dF}{d\theta} = -\frac{1}{2} \sqrt{\frac{1}{\sin\theta \cos^2 \theta}} \cos\theta (1 - 3\sin^2\theta)$$

It's easier to differentiate the other two. Anyway, I have:

$$\frac{dT}{T} = \frac{ds}{s} + \frac{1}{2}\frac{dx}{x} + \frac{(3\sin^2\theta - 1)\cos\theta}{2}d\theta$$

How can I express the last term in a $$d\theta / \theta$$ form? The only thing I can think of at the moment is using a small angle approximation, but I don't know how to justify that. And also, clearly, if I could use that, the small angle approximation for cosine has a square term.

2. Oct 24, 2009

### RoyalCat

For a function of multiple variables, $$T=f(x, s, \theta)$$, the error in the function, $$\Delta T=\sqrt{(\frac{\partial T}{\partial x}\cdot \Delta x)^2+(\frac{\partial T}{\partial s}\cdot \Delta s)^2+(\frac{\partial T}{\partial \theta}\cdot \Delta \theta)^2}$$

I suggest that you stop using the lowercase $$d$$ for the errors, because you're bound to get it mixed up with the derivatives you're taking.