Uncertainty with sine of angles?

1. Apr 27, 2013

Rampant

1. The problem statement, all variables and given/known data
I'm in a Year 10 Physics class, and we have been doing an experiment about Snell's Law (θincident = θrefracted). The experimental design is fairly simple: a beam of light (from a ray box with a single slit in front of it) is shone into a glass block. The angles of the incident and refracted rays with respect to the normal are measured with an ordinary protractor.

The uncertainty of the angle measurements is 0.5° (halving the smallest measurement). However, we are asked to graph the sine of the incident angle against the sine of the refracted angle. The graphing part and subsequent analysis is simple and I need no help with that - it's the uncertainty of the sin that I'm having difficulties with. How do I calculate the uncertainty of the sine of an angle?

2. Relevant equations

N/A

3. The attempt at a solution

I tried to find the uncertainty of the sine of the angle by finding the sine of the uncertainty of the angle. That was confusing, let me try again. I took the uncertainty of the angle and found the sine of that, but I'm pretty sure that's not correct.

2. Apr 27, 2013

CAF123

Generally, the uncertainty in a quantity is given by YOUR best estimate. The statements about the uncertainty being half the smallest division etc are only guidelines.

You will have to propagate your error. Consider the Taylor expansion: $$f(x+\Delta x) \approx f(x) + \frac{df}{dx}\Delta x$$ Then $$\Delta f \approx \frac{df}{dx}\Delta x$$

In your case, f = f(θ) = sinθ.

3. Apr 27, 2013

Staff: Mentor

Hi Rampant, Welcome to Physics Forums.

A pretty good estimate of the uncertainty can be found by evaluating the function at the max and min values of the given argument and then taking half the difference in values. In other words, in this case suppose that θ is the measured value and the uncertainty in the measurement is Δθ. Then:

$Δ = \left|\frac{sin(θ + Δθ) - sin(θ - Δθ)}{2}\right|$

should be a good estimate of the uncertainty in the sine of the angle θ.

EDIT: (I've assumed that your Year 10 physics class hasn't introduced calculus)

Last edited: Apr 27, 2013