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### Laws of Sines and Cosines

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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.
```

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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/
```
Associated Topics:
High School Trigonometry

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