Propagation of uncertainty with tangent function?

[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!

 

[scottdave]

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.

 

[Cheesycheese213]

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!

 

[haruspex]

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]

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!

 

[scottdave]

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 ).