Laws of Sines and Cosines

Date: 03/07/2002 at 22:18:11
From: Alison stroop
Subject: The laws of sines and cosines

Recently my math teacher assigned a problem in which the objective was 
to use the laws of sines and cosines.  Here are the specifics of 
triangle ABC: 

   Angle A is 23 degrees, the side opposite is x. 
   Angle B is y degrees, and the side opposite it is seven.  
   Angle C is z degrees, and the side opposite it is 5.  

I set up the problem using the law of cosines, and found that 
x = 3.0919. I then set up the rest of the problem using law of sines.  
However, when I figured the answer, the sine of 23 divided by 3.0919 
equals the sine of y over seven, the answer was approximately 62 
degrees. I also found that z = approximately 38.168. Obviously these 
threee angles didn't add up to 180 degrees.  

I figured the problem using cosines, and found that all the answers 
were the same except the measure of angle y. By using the law of 
cosines, I discovered that angle y = 118 degrees, approximately. When 
I figure using the law of cosines, all the angles in the triangle add 
up to 180 degrees.  I noticed that the answer I found for the measure 
of angle y using the law of sines was the supplement of the actual 
measure of angle y, using the law of cosines. Assuming that the 
results are consistent in all problems, why, if an angle is obtuse, 
does the law of sines find the supplement of the answer?  If this does 
not occur in all obtuse triangles, then why this one?  I am really  
confused, and rapidly losing faith in the laws.

Date: 03/07/2002 at 23:03:51
From: Doctor Peterson
Subject: Re: The laws of sines and cosines

Hi, Alison.

Just think about the nature of sines: the sine of, say, 70 degrees, is 
the same as the sine of 110 degrees (its supplement). Do you see why? 
Then any method that gives you the sine of an angle doesn't 
distinguish between that angle and its supplement; you have two 
choices for the angle, and have to use other information to decide 
which is correct. This is something like solving x^2 = 4 by taking the 
square root; if you just write x = sqrt(4), you have missed one of the 
two solutions. So the law of sines didn't give you the acute angle; 
you just forgot to consider the obtuse angle.

Our Dr. Math Trigonometry Formulas FAQ includes instructions for 
solving triangles:

http://mathforum.org/dr.math/faq/formulas/faq.trig.html#solveoblitri   

This is case III (SAS), and you are told to do just what you did to 
find x, but then to use the law of sines only for the angles opposite 
the two smallest sides, which have to be acute. Then you can find the 
third side by subtracting from 180 degrees, avoiding the need to 
decide whether to use the acute or obtuse angle with the given sine.

I hope that restores your faith!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/