The other trig teacher at my school came to me with a problem today. I, too, am stumped as to how to explain *why* the Law of Sines doesn't work to solve this problem correctly. After getting two answers I can see why, I just wouldn't have anticipated this issue a priori. I'm quite embarrassed to ask, but if anyone can offer help beyond "start with the Law of Cosines, stick with the Law of Cosines", I'm listening.
Here's the problem:
Now use the Law of Sines to solve for angle B, then calculate angle C.
Here's where the issue comes. Go back to the original triangle having solved for angle A, and instead of solving for angle B next, solve for C instead, and then calculate B.
Here's what I get. Solving for B first is in blue, solving for C first is in red:
There's only 1 right answer, and it's the blue one. The Law of Sines ratio doesn't work for all three pairs of angles/sides in the red triangle.
So the question is this: doing an arcsin won't give you the correct answer if your angle is in the 2nd quadrant (because the range of the arcsin function is (-90 degrees, 90 degrees). How would you know, then, to check for an obtuse angle in this case? Perhaps the better question is, without checking the Law of Sines ratio for all three pairs, would one know that the red answer is incorrect?
Have I done something procedurally incorrect? Again, any help is appreciated.