Determine the more accurate way of measuring the angle

Question

I need to measure an acute angle in a right triangle. There are $2$ methods.

1. Use a protractor with precision to $1^\circ$ to measure the angle.

The result is $5^\circ$.

2. Use a ruler with precision to $1\mathrm{ mm}$ to measure the opposite leg of the angle $a$ and the hypotenuse $c$, and calculate $\arcsin\frac ac$ with an sufficiently accurate calculator.

The result is $8.7\mathrm{ cm}$, $100\mathrm{ cm}$.

Determine the more accurate method to measure the angle. The criteria should be the percentage uncertainty.

For method $1$, $$\%U=\frac{1^\circ}{5^\circ}\times100\%=20\%.$$ I don't know how to find the percentage uncertainty of method $2$. Maybe the Taylor series should be used.

Answer 1

When working with uncertainty, I find it simpler to deal with ranges. I find that when I ask an uncertainty question about a range, it's typically rather easy to think through. One measures an angle as 5 degrees with a protractor whose precision is 1 degree. That's a range of 4 - 6 degrees.

(Or is it 4.5 - 5.5 degrees? Does 1 degree mean there's a one degree window of uncertainty, or that it's one degree in each direction? For this question it won't matter, as long as we define precision consistently. But for many real life questions, its wise to make sure both parties agree on how to notate precision. You'd be surprised how often a factor of two can creep in when using someone else's numbers. I will choose to interpret the precision numbers as +/- 1 degree and +/- 1mm, simply because that means I don't need to type as many digits)

Your ruler measurement is likewise 8.6 - 8.8cm. We'll use that range next.

What about the 100cm? Is that a measurement using the ruler, and thus actually 99.9 - 100.1cm? or is that a 100cm standard whose precision is so good that we can ignore it? Again, always ask. This one won't change the answer either, in this case, but it helps to think it through. I will choose to interpret it as a measurement because it makes it more obvious how to work with the uncertainty.

So we now have a range for $a$, 8.6 - 8.8cm, and a range for $c$, 99.9 - 100.1cm. We are going to divide $a$ by $c$ to get the argument that's going to be passed into $\sin^{-1}$. What's the range of values that division can take? On the low end, we would want the smallest numerator and the largest denominator: $\frac{8.6}{100.1}=0.0859140\ldots$. What' the largest value it can take? That would be the largest numerator and the smallest denominator: $\frac{8.8}{99.9}=0.088089\ldots$

Now that we have a range for the division, what is the range for the $\sin^{-1}$? I'll leave you to crunch the numbers for that. It's the same approach as the division: try to find the argument that would make $sin^{-1}$ big and the one that would make it small.

Now as written, you could probably stop with the ranges. They tell you which is more precise. But if they want you to do it as percentage uncertainty (to make sure you use a lesson taught in the class), you now have two ranges of angles (method 1 and method 2), and two measured values. Dividing the width of the range by the measured value will give you a percent error. No fancy calculus series' required!