Error Propagation Homework: Find T's Error

[bigevil]

Homework Statement

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

<br /> T = \sqrt{\frac{ks^2}{x \sin \theta \cos^2\theta}}<br />

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.

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.

 

[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.