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Angles in Trig Functions: Significant Figures


Date: 10/05/1999 at 01:07:53
From: Eric Prendergast
Subject: Significant figures in trig problems

What is the way to handle significant figures in angle measurements in 
trig problems?

For example, given a trail 1001 m long with an 11 degree grade above 
the horizontal, what is the elevation at the end of the trail?
(1001 m) * sin(11 degrees) = 190.9998 m or 190 m to two significant 
figures.

But, if the grade is 9 degrees...
(1001 m) * sin( 9 degrees) = 156.5909 m or 200 m to one significant 
figure.

The uncertainty in measuring 9 degrees versus 11 degrees doesn't seem 
large enough to justify changing from a level of one significant 
figure to a level of two significant figures. Are degrees and radians 
handled differently than other measurements?


Date: 10/05/1999 at 12:33:40
From: Doctor Peterson
Subject: Re: Significant figures in trig problems

Hi, Eric. Good question!

You're absolutely right in being suspicious about your significant 
figure calculation. But there's actually more involved here than you 
pointed out.

First, even if you were simply multiplying 1001 by 9 or 11, your 
comment about changing the number of significant figures is correct: 
significant figures are just a quick approximation, a "rule of thumb" 
for working with uncertainty in measurement, and you'll sometimes have 
more or less accuracy than your figures indicate. The best way to deal 
with precision is to write numbers with a range of error attached: 
1001 +- 0.5, for example. If you really care about precision, this is 
the only way to go.

Second, the sine function has an effect you haven't considered. Moving 
from 11 degrees to, say, 11.5 degrees slides you along the sine curve, 
changing the range of error. When you learn calculus, you'll find out 
how to determine the slope of the sine curve. It turns out that this 
slope is equal to the cosine, if you are using radians. So as far as 
error is concerned, taking the sine of 11 degrees can be thought of as 
if it were multiplication by pi/180 (to convert degrees to radians) 
followed by multiplication by cos(11) = 0.982.

So if your numbers are really 1001 +- 0.5 and 11 +- 0.5 degrees, your 
calculation is

     (1001 +- 0.5)(sin(11 +- 0.5 deg)) =

     (1001 +- 0.5)(sin(11) +- 0.5 * pi/180 * cos(11)) =

     (1001 +- 0.5)(sin(11) +- 0.5 * 0.0175 * 0.982) =

     (1001 +- 0.5)(sin(11) +- 0.0086) =

     1001 sin(11) +- 0.5 sin(11) +- 1001 * 0.0086 +- 0.5 * 0.0086 =

     191 +- 0.095 +- 8.58 +- 0.0043

The 8.58 (which comes from 1001 times the cosine term) overwhelms the 
other error terms (that's why we keep the fewest significant digits in 
a multiplication), so our answer is

     191 +- 9

This means you really have less than one significant digit, but you 
are justified in writing the answer as 190, with two significant 
digits. (This agrees exactly with your estimate that it is either one 
or two figures.) In this case, because the cosine is large, the fact 
that you are taking the sine has little effect on the precision; but 
if the cosine were small (for angles around 90 degrees), you would 
find you had much more precision than you thought you did, because a 
change in the angle would make only a small change in the value of the 
sine.

I don't know of a good rule for significant figures in working with 
trig functions; usually something like my approach with errors is 
used. If you want to check the accuracy of your answers, you could 
skip all this and just repeat your calculation with, say, 1001.5  
sin(11.5) and see how far off it is. That can give you a good feel for 
how the error really works.

I've probably written far too much to say far too little; if you don't 
follow any of this and would like to discuss it more, please write 
back. If you're interested in pursuing this, here are a few sites I've 
found that explore these issues:

 Significant Figures and Rounding Rules - Christopher Mulliss
 http://www.angelfire.com/oh/cmulliss/   

 Fundamentals of Data Analysis in Physical Sciences - Tatiana Allen
 http://www.utc.edu/~tbilgild/fundofda.html   

 Error Analysis - Frank Wolfs
 http://teacher.nsrl.rochester.edu/Phy_labs/AppendixB/AppendixB.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Trigonometry

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