# Propagation of uncertainty with tangent function?

• Cheesycheese213
In summary, the lab's uncertainty in their calculated refractive index was due to the software they were using to graph the data not being defaulted to radians.
Cheesycheese213
Homework Statement
Calculate the uncertainty of a value (n) calculated by the equation n=1.0003tanθ
Relevant Equations
(shown below)
For a lab, I needed to calculate the uncertainty of a refractive index that was found using Snell's law. I found an equation online for propagation of error for any general function, which was

I thought that since my equation was

I could just get rid of the variable y, and have

After inputting my values θ = 53.61 and Δθ = 0.1, I got
Δn = 0.284193...
Which I was sort of confused by? My value for n itself is only around 1.357, and I wasn't sure why the uncertainty was so large in comparison. I was wondering if I did something wrong?

For more context, I am calculating n for many different angles, and when graphing the values either the error bars look really big, or the slope is almost insignificant from the scale.

Thanks!

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Try using Radians for your angle measurement, instead of degrees. That's the only way you can take derivatives of trig functions.

Also, depending on the software package you use to graph, it could be defaulting to radians.

sysprog and Cheesycheese213
scottdave said:
Try using Radians for your angle measurement, instead of degrees. That's the only way you can take derivatives of trig functions.

Also, depending on the software package you use to graph, it could be defaulting to radians.
Ohhh thank you so much!
I just tried converting everything to radians, and I got a much smaller number (0.004961...). Its really interesting how it needs radians over degrees :D Thank you again!

sysprog, jim mcnamara and scottdave
Cheesycheese213 said:
interesting how it needs radians over degrees
It's not so much that it has to be radians. Rather, it is because the standard derivatives we learn, like d sin(x)/dx = cos (x), assume x is in radians.

Cheesycheese213, scottdave and sysprog
haruspex said:
It's not so much that it has to be radians. Rather, it is because the standard derivatives we learn, like d sin(x)/dx = cos (x), assume x is in radians.

Ohhhh I see! I didn't connect that! Thank you!

Yes, like @haruspex said, you could have a scale factor of (π radians)/180° inside the sin(x), so taking the derivative of sin(x) with x in degrees becomes (π/180)*cos(x).

This means the sine would change approx 0.01745 vertically for every Δx = 1° ( when you are near x = 0 ).

## 1. What is the tangent function and why is it important in uncertainty propagation?

The tangent function is a mathematical function that relates the ratio of the opposite side to the adjacent side of a right triangle. It is important in uncertainty propagation because it is used to calculate uncertainties in trigonometric functions, which are commonly used in scientific calculations.

## 2. How is uncertainty propagated with the tangent function?

Uncertainty is propagated with the tangent function by using the formula for the propagation of uncertainty, which involves taking the partial derivative of the tangent function with respect to the variables involved, and multiplying it by the uncertainties of those variables.

## 3. Can the tangent function be used to propagate uncertainties in other functions?

Yes, the tangent function can be used to propagate uncertainties in other trigonometric functions, such as sine and cosine, as well as in more complex functions that involve trigonometric functions.

## 4. Are there any limitations to using the tangent function for uncertainty propagation?

One limitation of using the tangent function for uncertainty propagation is that it assumes that the uncertainties in the variables involved are small. If the uncertainties are large, the tangent function may not provide an accurate estimation of the propagated uncertainty.

## 5. How can I calculate uncertainties in the tangent function?

To calculate uncertainties in the tangent function, you will need to know the uncertainties of the variables involved and use the formula for uncertainty propagation. You can also use software or online calculators that have built-in functions for uncertainty propagation with the tangent function.

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