Breaking the Sine Law

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.