# Calculating Uncertainty for a Chain of Trig Functions

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1. Oct 22, 2016

### Ryan Hardt

1. The problem statement, all variables and given/known data
I have a series of 12 values that I need to calculate the Theoretical Intensity, I, using the formula below.

I have found values for all variables and their uncertainties, and have calculated the I value for each set using the formula. Now I need to calculate the uncertainty.

Example Values:
I0 = 1.80 ± 0.01
θ = 0.17 ± 0.02 rad
λ = 0.0371 ± 0.0026 m
d = 0.072 ± 0.001 m

And I arrived at an I value of 0.043.

2. Relevant equations

I = I0cos2((π/λ)d*sin(θ))

3. The attempt at a solution
I know the basic formulas for calculating uncertainties of addition, multiplication, and power functions in quadrature. I don't fully understand how to calculate uncertainty or trig functions, but I assume you just take the function of the value and uncertainty inside quadrature? As for a chain of multiple functions (let alone trigonometric) I am stuck, and I could not find documentation on it easily.

Thanks

2. Oct 22, 2016

### robphy

3. Oct 22, 2016

### Staff: Mentor

Hi Ryan, Welcome to Physics Forums.

The usual approach for propagating uncertainties when dealing with a general function of multiple variables is to use a bit of calculus to determine the individual contributions of the uncertainties and adding them in quadrature. The calculus here involves partial derivatives of the overall function with respect to each of the variables that have an uncertainty. If you know the necessary calculus then that would be how to proceed.

For example, suppose you have some function of three variables f(x,y,z), where each of x, y, and z has some associated uncertainty Δx, Δy, Δz. Then the propagated uncertainty Δf would be given by:

$$Δf = \sqrt{ \left( \frac{\partial f}{\partial x} \right)^2 Δx^2 + \left( \frac{\partial f}{\partial y} \right)^2 Δy^2 + \left( \frac{\partial f}{\partial z} \right)^2 Δz^2}$$

4. Oct 22, 2016

### Ryan Hardt

I actually saw that page while I was searching for documentation and must have missed over the general rules to use derivations. Thank you @robphy.

And thank you @gneill, I do have the background and will take said derivatives. I will post a reply again once I am able to work it out tomorrow, and I hope I can get some check that I have done it right or am on the right track.