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### Ambiguous Cases - Laws of Cosines and Sines

```
Date: 04/26/2000 at 14:36:43
From: Kim Feinman
Subject: Are there ambiguous cases with the Law of Cosines?

I am working on solving triangles using the law of cosines and have
run into a problem where I get different solutions, and I don't
understand why.

Let's say I have triangle ABC, with sides of length a = 3.2, b = 4.3,
and c = 5.1. I start off using the law of cosines and I find the
measure of angle C is 84 degrees. Then I use the law of sines and find
angle A = 39 degrees.

Here is where I run into trouble. If I just subtract the two angles
from 180, I get angle B = 57 degrees, but if I go ahead and use the
law of sines with sides b and c, I get angle B = 33 degrees.

Why is this? Is there an ambiguous case? Because if I try to find the
measure of angle B using the law of sines using sides a and b,
instead of b and c as I did, I get the 57 degrees I got earlier just
by subtracting. Shouldn't both methods work, regardless of which
angles and sides I use? I think this relates to an ambiguous case of
the law of sines, but I don't understand why.

Thanks so much!
Kim
```

```
Date: 04/26/2000 at 23:06:29
From: Doctor Peterson
Subject: Re: Are there ambiguous cases with the Law of Cosines?

Hi, Kim.

It sounds to me like you just took the inverse cosine when you meant
to take the inverse sine. That would give you the complement of the
right answer. Let me know if that's not the explanation.

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

```
Date: 04/27/2000 at 07:23:03
From: Kim Feinman
Subject: Re: Are there ambiguous cases with the Law of Cosines?

Hi,

No, when I use the inverse sin I still get the 37.

[3.2*sin(84)]/5.1 = sin A
so
sin^-1 [3.2*sin(84)/5.1] = 37

Thanks again,
Kim
```

```
Date: 04/27/2000 at 08:49:44
From: Doctor Peterson
Subject: Re: Are there ambiguous cases with the Law of Cosines?

Hi, Kim.

As I understand it, your problem was not in calculating A, but B. Your
calculation gives A = 38 degrees (approximately); then we find B as

sin(B) = sin(C) * 4.3/5.1 = 0.8385

B = sin^-1(0.8385) = 57 degrees

You said you got B = 33 degrees using the law of sines, which is the
complement of this correct answer. Please check your calculations at
this step and see whether I'm right that you used the inverse cosine
here.

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

```
Date: 04/28/2000 at 12:49:40
From: Kim Feinman
Subject: Re: are there ambiguous cases with the Law of Cosines?

Hi again,

Okay. I am _really_ struggling.

In the example you emailed me, I promise I am not using the inverse
cosine. In my calculator I type [2nd] [sin], which is the sin^-1
function, right?

Here is another case. Triangle ABC has sides a = 126 and b = 80.1.
Angle C is 39.4 degrees.

So I use the law of cosines to find that side c is 81.8.

Then I use the law of sines with side/angles a and c and I come up
with an angle A of 77.88 degrees.

When I subtract 77.88 and 39.4 from 180, I have an angle B that has to
be 62.8 degrees.

BUT if I use the law of sines with angles/sides b and c, I come up
with angle of of 38.4 degrees, and so by subtracting angle A has to be
102.2 degrees. This is the answer in the book.

What is happening? I really just do NOT understand why I am coming up
with two different answers using it the two different paths.

Thank you!
-Kim
```

```
Date: 04/28/2000 at 13:35:01
From: Doctor Peterson
Subject: Re: are there ambiguous cases with the Law of Cosines?

Hi, Kim. Thanks for writing back and showing your work in detail.
That's what we've been needing!

In your original problem, I get the same result for B whether I apply
the law of sines to A and B or to B and C. I still feel that you just
did a calculation wrong. The triangle is acute, so nothing is
ambiguous.

Have you really been recalculating that specific problem to verify
that it comes out wrong even though you use the inverse sine, or have
you been assuming that you had the same error in several different
problems, and checking your work on a different one? It's important to
work on one problem at a time so we don't get confused.

But this new problem is different. It DOES involve an ambiguous sine.
Angle A could be either 77.88 or 102.12 degrees in order for sin(A) to
be 0.977. You have to determine which angles are acute by comparing
the three sides; since b and c are the smallest sides, angles B and C
must be acute. Therefore, you should first calculate B and choose the
acute value, then determine A from that. You may want to look at our
FAQ on solving triangles at:

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

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

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