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### Angles in Trig Functions: Significant Figures

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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!

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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