# propagation of error - sin^4

I am currently working on an analysis of a Rutherford scattering and encountered a somehow strange behaviour for the errors. It basically boils down to the behaviour of: $$\sin(\theta/2)^4$$ For simplicity, I am assuming I have just a function: $$y(\theta)=\sin(\theta/2)^4$$ One point in my data is the angle: $$\theta=66\pm4$$. Now i want to calculate $y(66)$ and its error $\Delta y(66)$ The first one is simply the function: $y(66)=\sin(33)^4=0.08799...$ Now i want to calculate the error via propagation of error: $$\Delta f(\theta) = \frac{\partial f }{\partial \theta}=\cos(\theta/2)\sin(\theta/2)^3 2\Delta \theta$$ If I plug in my value and its error from above I am getting $\Delta f(66\pm4)=1.0839...$

This means my error is two whole magnitudes larger than my value, althogh the initial error 4 is one magnitude lower than the value 66. How can this be? Am I doing something wrong or is there some kind of trick I have to use?

## 1 Answer

The trick is to convert to radians. You're mixing radians and degrees, which I think makes your error too big by $180/\pi$.

• Well, sometimes it is just the classic mistakes... I converted the argument of the sin and cos intro radians but totally forgot to transform $\Delta \theta$ as well. Thank you for that. – Djoser42 Jun 22 '16 at 20:54
• Well, not a trick per se so much as a realization that the differential form the OP used is not correct for measuring angles in degrees rather than the natural units of radians... – Jon Custer Jun 22 '16 at 20:54