It all started yesterday. I was checking homework from my grade 10 class and saw different answers to the same question. I had found the diagram on-line on mathisfun.com. Ironically, when I "steal" questions like this, I do check for the ambiguous case as it is not part of the grade 10 curriculum. Clearly I didn't do this very well! Here was the diagram - I had asked my students to solve the triangle.
Once I realized that something was up, I triple-checked my work calculating the length of side c, then found angles A and B with both the sine law and cosine law. I could see that for angle B I was getting the first and second quadrant answers, but because I had all three side lengths of the triangle, I could not see the ambiguous case. Nope. No way. What was I missing, I asked? Literally.
As always, the MTBoS came through in spades! Look at all these amazing replies!
Although I could understand the explanations, it wasn't until I saw the diagrams from Sean Sweeney that my brain clicked that for this part of the question, it was the ambiguous case because I wasn't using the third side. I don't know why I couldn't see that before, but I felt like such an idiot. I mean, I teach trig in grade 10 and in grade 12. I know trig. I get trig. I love trig. Why couldn't I figure this out on my own? I haven't taught ambiguous case for years and years (it's in grade 11 in our curriculum, which I never seem to teach), but still, I have taught it and really do understand it. I think the purpose of this post, along with singing the praises and thanking the awesome folks on Twitter, is to remind myself (and others?) that it's okay to not know all the answers all the time. It's okay to ask questions. I likely don't look stupid for asking the questions, despite feeling that way. I certainly never think that of others when they ask questions, so why does that not apply to me? As it turns out, I now know more about the ambiguous case than I think I ever have! Mike Lawler even wrote a cool blog post (link here) inspired (that seems a bit of a strong word but I can't come up with a better one) by my original tweet:
Thanks to everyone who helped me think this through. I greatly appreciate it.
I started with this picture and explained that the surveyor was trying to determine the width of the river (if someone can give me the source of this question I would greatly appreciate being able to give credit):
They then worked in groups to find the width of the river using this diagram (it was so nice to see and hear them talking math!). Most groups started by finding the missing angle in the triangle and then got stuck. They knew they needed to use trigonometry but either tried with this triangle as-is, or realized that they didn't have a right angle. I circulated and asked them what they needed to use trig and each group replied that they needed a right triangle. So I asked how they could create one and walked away.
Many groups added altitudes that were not helpful as they didn't know a side length of the right triangle they had created. I asked if they could create a different right triangle (and walked away).
Here is the (edited) solution of the group that didn't need any hints from me:
I told them that they had solved for a side in a non-right triangle! I asked if they would like to not have to always draw the triangle and add the altitude and so on. This led us to...
We talked about when each form would be useful and what information you needed to use the sine law before working on some examples. Many had a really hard time with the first example. I tried to help guide them by writing the sine law on the whiteboard and putting check marks next the information we knew so they could see that we only needed the proportion using A & B.
The next example required them to notice that they didn't have an angle-side pair so needed to find the third angle before using the sine law.
In the final example they found the measure of an angle.
Here is today's homework.
