Law of Sines--Ambiguous Case

Teaching this particular topic in the past has created numerous headaches for both me and my students.

For those of you who need a reminder, the ambiguous case occurs when one uses the law of sines to determine missing measures of a triangle when given two sides and an angle opposite one of those angles (SSA). In this ambiguous case, three possible situations can occur: 1) no triangle with the given information exists, 2) one such triangle exists, or 3) two distinct triangles may be formed that satisfy the given conditions. These possibilities are summarized in the diagrams below:

Suppose we are given side a, side b and angle A of triangle ABC. Let h equal the height of the "triangle". Note that h can be found using h = b(sina). (Do you see why?)

If angle A is acute, and a < h, no such triangle exists. If angle A is acute, and a = h, one possible triangle exists. If angle A is acute, and a > b, one possible triangle exists. If angle A is acute, and h < a < b, two possible triangles exist. If angle A is obtuse, and a < b or a = b, no such triangle exists. If angle A is obtuse, and a > b, one such triangle exists. As I previously stated, this has been a difficult topic to get across to students in the past. (To be honest, I remember having similar difficulties the first few times I was exposed to the Law of Sines.) I tried various techniques, but most students seemed to have a hard time visualizing the different cases.

The link below will take you to a GSP 4.0 demonstration related to this topic that it similar to one I used this past year while teaching an Euclidean Geometry course at Athens Christian School. After the presentation, students seemed much more comfortable with the ideas when compared to students from previous years. A larger percentage of students also seemed to handle problems related to the ambiguous case better then their predecessors. (I did not actually go back and compare grades; this observation is based solely on my memory.)

Click on the link below to see the demonstration (GSP 4.0 required). At the top.

GSP demo

I presented this demonstration simply by using the previous sketch to illustrate each case. Students could also use this demo to explore different given information and arrive at the generalizations on their own. A more ambitious project would be to allow students to generate their own GSP presentation. Individual teachers would have to weigh such factors as student familiarity with GSP and time constraints in completing such a project.
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