# Calculating Uncertainty With A Sine Function

1. Dec 2, 2011

### sunjay03

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

I am doing this calculation in my lab:
h = sin(24.0°)[(180.0cm)(1m/100cm)]

The uncertainty on the angle is ±0.5° and on the length it is also ±0.5cm. How can I go about calculating the uncertainty? If you know the answer, do you mind putting it in terms of a non-calculus student. I googled and found this, but could not understand a word of what they meant:
http://www.sosmath.com/CBB/viewtopic.php?t=45581

I'm in more need of a simple formula I can plug numbers into to get the correct uncertainty. The understanding of the math behind it is not as relevant at this time.

2. Relevant equations

N/A

3. The attempt at a solution

I can get the answer, but not the uncertainty. Here is the answer:
h = 0.732m

2. Dec 2, 2011

### Staff: Mentor

Presumably the factor (1m/100cm) is simply a conversion factor that has infinite precision. You have only two variables with uncertainties attached, namely θ and x, where:

h(θ,x) = sin(θ)*x*(1m/100cm)

is the function that returns your result, and for which you want to propagate the uncertainties of the variables.

Unfortunately, the way to compute the uncertainty in this situation does involve a bit of calculus. Have you had any calculus instruction at all?

3. Dec 2, 2011

### sunjay03

Not at all. Is there no formula you can simply give me to calculate the uncertainty given the function you just wrote out?

4. Dec 2, 2011

### Staff: Mentor

It's strictly against Forum policy to just give out answers; the student has to do the work. We're here to advise, give hints, spot errors, and so forth.

Perhaps you can explain your situation a bit. What course is lab for? Is it usual to have assignments that require knowledge that the student hasn't yet acquired?

5. Dec 2, 2011

### sunjay03

This is for my IB Group IV Project. I myself am at a Physics 20 level with Math 30 (Alberta curriculum) experience. The Group IV Project is a large group project which involves 1 lab in each of the three subjects: Physics, Biology and Chemistry. These labs are student directed and thus designed and executed by the students.

Normally, I would not need to calculate the uncertainty when it has to do with sine functions, and thus do not have the background or know-how to calculate this on my own. That is why I just wanted some equation which gives me the uncertainty I want. The math behind the calculation is not relevant to my understanding since I am not required to know how to do it at all. I simply want to get my lab done and need the correct uncertainty formula for this situation. You need not give me the answer, how about just a formula that allows me to find the answer myself?

Thank you

6. Dec 2, 2011

### Staff: Mentor

I see. Well, I can give you a description of the general procedure. You'll have to follow up with a bit of investigation to make it work.

When you have a function of several variables f(x,y,...), where each variable has some uncertainty associated with it, Δx,Δy,..., then the procedure is:

For each variable:
1. Take the partial derivative of the function with respect the variable.
2. Plug in the measured values for all variables into the partial derivative.
3. Multiply the partial derivative by the uncertainty associated with the particular variable.
4. Square the result

Then take the square root of the sum of the values obtained. In symbols for two variables, given a function f(x,y) then:

$$\Delta f = \sqrt{\left(\frac{\partial f(x,y)}{\partial x}\Delta x\right)^2 + \left(\frac{\partial f(x,y)}{\partial y}\Delta y\right)^2}$$

A partial derivative is where you treat only one variable as a variable and treat all the others as constants. Essentially this finds the individual variation in the function with respect to each variable at a given point on the function. Multiply this "sensitivity to change" by the size of the error for that variable and it tells you how much the function is expected to vary as a result of that error. These individual variations are added in quadrature (square root of the sum of the squares, just like for vector components).

Further information on error analysis can be found here.

In order to apply this procedure you'll need to find out how to take the derivative of your function (yes, it's calculus). I might mention that if you do a web search you might just find online applications that will differentiate an expression. Search for "online derivative calculator".

7. Dec 2, 2011

### Delphi51

I hate to butt in, but can't resist because I'm a veteran of many years teaching Alberta physics 20. At that level, we don't know how to do sophisticated error propagation. If we have a number like 10 ± 1 (ten with an error estimate of 1), we just run through the whole calculation with 10 (best value) and again with 9 or 11 (low or high value). Say we get 100, 95 and 106 for those calcs. Then we say the answer is 100 ± 6. This actually works out almost exactly to what you get with the calculus formulas and is quite understandable. I find it amazing that many students make it to university without ever having done any error calculations (other than how many % my answer is off the "right" answer) when it is so easy to do it this way!

In your case you have two inputs multiplied, so you just use two "lows" to get the "low" answer. If you had dividing, you would use a "low" for the numerator and a "high" for the denominator to get the "low" answer.

8. Dec 2, 2011

### Staff: Mentor

Well, I for one won't argue with a veteran If that's the way it's done for that course then that's the way to go. I do wonder why the OP wasn't familiar with the accepted method.

9. Dec 2, 2011

### Delphi51

Thanks, g. The answer is I was a bit of a rebel - there is NO sensible error treatment in the course but I thought it was essential for students to get a taste of it. There is quite a bit of "finding the formula" in the course. Say you are trying to find Newton's F = ma and you measure your F's and a's, graph them and find it is a reasonably straight line. But not exactly. I taught that you can't reach any conclusion unless you have error estimates. And you can never prove any physics formula exactly; the best you can say is that it fits the data to within the experimental error. The OP mentioned he is doing an "AP" course - that is "Advanced Placement" - he could well be on his way to a university honours program and end up finding new physics formulas for us. He or she needs to know something about experimental error and has at last found a teacher requiring it.

I fear error calcs may be slipping away from first year physics at university here because calculus physics is now postponed to second year.

10. Dec 2, 2011

### Staff: Mentor

Gaahhh! Nooooo!

Should make for some "interesting" questions in the forums when students are puzzled by their wiggly line graphs.