Law of sines: Unambiguous case

In summary, the conversation discusses the ambiguity in defining a triangle given two sides and the intermediate angle. The problem is solved using the Law of Cosines and Law of Sines, resulting in two possible triangles. The question of how to resolve this issue is also raised, with one approach being to check for both possible values of each unknown angle using the sine identity, and then finding the combination that results in a sum of 180 degrees.
  • #1
SweatingBear
119
0
Hello again, forum.

Is it not true that a triangle is unambiguously defined given two sides and the intermediate angle? That is at least what I learned from studying congruence in geometry. Here's the problem: We have the triangle below (the picture is given in the problem and I have redrawn it) and are asked to find the measurement of the other two angles and the side BC.

9uvDM62.png


Now, intuitively, there is only one possible length for BC for which a triangle can be constructed. The angle at C is fixed because changing that very angle would change the length of BA (which is given and thus constrained). Similar arguments apply to angle B.

From these arguments, it is reasonable to expect only one possible triangle from the given data. To start off we can use law of cosines to determine |BC| and from thereon make use law of sines.

The problem though is when I apply the law of sines in order to compute the other two angles, I am able to form two different triangles with different measurements of their angles.

Triangle #1: \(\displaystyle \angle A = 23.8^\circ\), \(\displaystyle \angle B = 34.3^\circ\) and \(\displaystyle \angle C = 121.9^\circ\).

triangle 23.8°,34.3°,121.9° - Wolfram|Alpha

Triangle #2: \(\displaystyle \angle A = 23.8^\circ\), \(\displaystyle \angle B = 145.7^\circ\) and \(\displaystyle \angle C = 10.5^\circ\).

triangle 10.5°,23.8°,145.7° - Wolfram|Alpha

Now this is very strange, I have not come across a triangle (with two given sides and the intermediate angle) which is not unambiguously defined until now. How is this even possible? Clearly in triangle #2, it is not possible for the angle at B to equal 145.7 degrees since this would change the length of AC. But the algebra does not "see" that, simply due to \(\displaystyle \sin (180^\circ - v) \equiv \sin (v)\).

So, forum, how do we resolve this issue? Help much appreciated!
 
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  • #2
I would first use the Law of Cosines to determine:

\(\displaystyle \overline{BC}\approx4.264311887306\)

Then using the Law of Sines, I find:

\(\displaystyle \angle C\approx34.6^{\circ},\,145.4^{\circ}\)

\(\displaystyle \angle B\approx58.4^{\circ},\,121.6^{\circ}\)

The only combination which has the sum of the thee angles as \(\displaystyle 180^{\circ}\) is:

\(\displaystyle \angle A=23.8^{\circ},\,\angle B\approx121.6^{\circ},\,\angle C\approx34.6^{\circ}\)
 
  • #3
Thanks for the reply!

The only difference between my and your approach is that you calculated every angle in the triangle using law of sines, whereas I calculated only one angle and the final one assuming the angle sum is 180. Why exactly did my approach fail in this case? I have had no issues with it before until this very problem.
 
  • #4
I was taught, in this case, to look for both possible values of each of the two unknown angles via the sine identity you cited, and only then to look for that combination of angles who sum is $180^{\circ}$.
 
  • #5
Seemingly a much better approach than mine. Thanks a heap!
 

Related to Law of sines: Unambiguous case

What is the "Law of sines: Unambiguous case"?

The Law of sines is a mathematical formula used to find the missing angles or sides of a triangle when given at least three of its sides and/or angles. The unambiguous case refers to the situation when there is only one possible solution for the triangle, meaning there are no ambiguous or multiple solutions.

When is the Law of sines used?

The Law of sines is used when working with triangles that are not right triangles, meaning they do not have a 90-degree angle. It is commonly used in trigonometry and geometry to solve problems involving triangles.

What are the three parts of the Law of sines?

The three parts of the Law of sines are the sine of an angle divided by its opposite side, which is equal for all angles and sides in a triangle. This can be expressed as sin(A)/a = sin(B)/b = sin(C)/c, where A, B, and C are the angles of the triangle and a, b, and c are the opposite sides.

How is the Law of sines derived?

The Law of sines is derived from the Law of cosines, which is a formula used to find the length of a side of a triangle when given the lengths of the other two sides and the angle in between them. By manipulating the Law of cosines, we can derive the Law of sines.

Are there any limitations to using the Law of sines?

Yes, the Law of sines can only be used in triangles that have three known parts, including at least one side and its opposite angle. Additionally, it can only be used to find one missing part at a time, meaning multiple applications may be needed to solve a more complex problem. Moreover, the Law of sines only works for non-right triangles, so it cannot be used for right triangles.

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