Tuesday, 8 July 2014

Unambiguous Sine Law


Sine Law. In any triangle, it states that the proportion between a side, and the sine of each corresponding angle, will be the same.



There's no problem with it in Grade 10, when all angles are acute... but as soon as obtuse angles are possible, there is something of a problem. Because inverse sine isn't a function, and a calculator will only return a single acute answer - which may not be the only possible answer, or may be completely wrong.



Here is a (perhaps non-traditional) way of dealing with it. I suggest "non-traditional" as this focusses on the triangle side correspondence, not the angles.


ONE PHRASE


"The longest side of a triangle is across from the largest angle."

That's a simple mathematical truth. Presumably one can see that the only way there ISN'T a longest side is if angles are equal. Now, allow me to apply that phrase to the various information cases when dealing with a triangle:
1) Three Sides (SSS): Unambiguous. You are given the longest side, and know it's length. (If you want, you can find an angle with cosine law.)
2) Three Angles (AAA): Unsolvable. You know where the longest side is, but cannot determine lengths.
3) Two Angles, Side (AAS): Unambiguous. See (2), now with the added knowledge of a length. (Solved with sine law.)

Math classes often split the last case, Two Sides, Angle, into two:
Sine or Cosine?
4) Two Sides, Angle Contained (SAS). Unambiguous. We do not know whether the missing side is the longest (unless the angle is obtuse!), but the angle DOES have clearly defined segments. So we can figure out the missing length (with cosine law) if we wish to know for certain.

5) Two Sides, Angle Outside (SSA). No clear segments. Breaks down as follows:
a. Unambiguous if, like (4), the angle is obtuse -- as we know the corresponding side is the longest.

b. Unambiguous if that corresponding side (the one in proportion with the angle) is long. This takes a bit of reasoning but no real mathematics - follow this logic.


Summarizing the "inverse sine" problem.
As the corresponding side is longer than the other given side, either it IS the longest, or it is the second longest. (It cannot be the shortest.) So there are two options... isn't that ambiguous? Except the angle we now solve for (with Sine Law) must ALSO be acute. Regardless of the outcome, this presents us with case (3) above. Meaning the sum of those angles (whatever they are) will resolve the hint of ambiguity.

The only case left is 5c. The angle is acute, but the corresponding side is short. So you don't know if it IS the shortest... or the second shortest. Meaning you don't know if the other given side happens to be the second largest... or the largest. This ambiguity CANNOT be resolved, because upon solving the angle it does NOT have to be acute. Hence both cases (the angle and the angle's supplement) must be considered. Which is the "Ambiguous Case", owing to the dysfunctional nature of inverse sine.


One could argue there is another case - the triangle is impossible. But hopefully you'll notice this in 5a, if the corresponding side is given as NOT being the longest. Similarly, it will fall out naturally when an attempt to solve 5c occurs; the given angle/side correspondence produces an impossible ratio for sine. The shortest side in the given information is simply too short. (The case of a single triangle of 90 degrees also falls out, as the supplement to 90 is also 90.)


CLASSROOM APPLICATION


I don't recommend breaking down all those cases for students. I certainly don't do it - I even found it hard to explain myself above. What is probably best is if they explore the possibilities on their own. I can't speak to that - I'm not quite that free spirited yet.

What I do is start with the One Phrase at the top, and then start tossing sides and angles at them. I even admit to keeping some structure in that - the cases on the first day deal with the Sine Law (AAS, SSA) and the ones on the second day with the Cosine Law (SAS, SSS). Though I don't actually categorize things like 'SAS' until the whole thing is done.


Traditional model?
It feels like it goes better than a traditional exploration of "the ambiguous case", where one creates a sketch and asks whether the opposing side can be "swung around" to connect as a second triangle. It also shows why, given SSS, one should find the largest angle first, to avoid accidentally generating one's own "ambiguous case" (when the triangle is in no way ambiguous). Granted, it's not as in depth as using given SSA information to calculate the triangle height, using that as a point of comparison, but that can be done as an extension.

Problems? Some students felt that SAS could be impossible, despite finding the missing side, because angles/sides didn't synch up. (Creation of the artificial "ambiguous case".) There was also a tendency to say "impossible" if an invalid ratio occurred - not due to the situation, but due to incorrect mathematics. Finally, there were acute answers given when there was ONLY an obtuse angle in a clearly drawn triangle... though it's unclear whether that was belief in "no ambiguity" meaning "must be acute", or merely not thinking the problem through.

Either way, I feel using this angle-side relationship extends somewhat naturally from the acute work in Grade 10. I welcome any additional thoughts on the matter.


ONE MORE THING



Focusing on lengths instead of angles might cause students to use the sine law even in right triangles... I honestly don't know, because they have a tendency to do that anyway. Poor 'sin/cos/tan'. To me, neglecting the basic ratios is akin to grabbing a hammer every time you need to hang a picture - you can do it, but if you've got a screw in your other hand, that's not really what the hammer was designed for. (One wonders then why students don't use the Cosine Law instead of the Pythagorean Theorem.) Feel free to enlighten me here too.

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