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/
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