- #1

zexxa

- 32

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## Homework Statement

Hi there, the issue I'm having right now is finding the right way to calculate uncertainties when trigonometric terms come in play.

The question goes like this, we're given 2 sets of 5 angles in degrees, the incident angle (10.0, 20.0, 30.0, 40.0, 50.0) and the other is the refracted angle (6.9, 13.6, 20.0, 25.4, 31.1). The measurements were taken with an analogue protractor with a ±0.1° resolution.

The refractive index

*n*is to be calculated with it's standard uncertainty.

## Homework Equations

[tex]

n = \frac{sin(α)} {sin(β)}

[/tex][tex]

dn = \sqrt { (\frac {δn} {δα} dα)^2 + (\frac {δn} {δβ} dβ)^2}

[/tex]

## The Attempt at a Solution

I first tabulated the data and found

*n*for each α,β entry and averaged all 5 values of

*n*which was 1.466461.

The standard deviation of

*n*is 0.024073, which then I used to find its Type A uncertainty with the equation [tex]u=\frac{σ}{\sqrt 5}[/tex]

When I tried to find the Type B uncertainty however, I hit a wall. This was rest of my work:

[tex]

\frac {δn}{δα}=\frac {cos(α)} {sin(β)}

[/tex][tex]

\frac {δn}{δβ}=\frac {-sin(α) cos(β)} {sin(β)^2}

[/tex][tex]

∴dn= \sqrt {n^2 (cot(β)^2 dβ^2 -dα^2) + \frac{dα^2}{sin(β)^2}}

[/tex]

I do not know what I should substitute in place of β when calculating since the value of β is a dependent variable in the experiment and have been at a loss for a few days.

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