Lesson 8.6
The Chord-Sine Connection
The Law of Sines was a circle theorem all along
This is the payoff, not just of the unit but arguably of the last three. In Unit 7 students derived the law of sines and used it fluently, but one question was left quietly open, perhaps without ever being named as open: that common ratio, the thing a/sin A and b/sin B and c/sin C all equal, what is it? Today they find out that it is 2R, the diameter of the circle that passes through all three vertices, and that the law of sines was a theorem about circles the whole time.
The most satisfying part was the very start. Every group constructed the circumscribed circle around their triangle using the chord-bisector theorem from 8.5, with no guidance from me. The tool had become theirs. From there the argument is the inscribed angle theorem plus Thales’ theorem, assembled into a single diagram.
When groups moved to measuring and conjecturing, I did steer them to write the law of sines in the a/sin A form rather than the reciprocal sin A / a. I think that is a fair place to point: the ratio you want them to recognize as the diameter is sitting right there in that arrangement. With that nudge, groups conjectured that the ratio equaled the diameter without much trouble.
The proof is where it got heavy. I gave groups the full picture with both auxiliary constructions, and even then it was a big ask, most assembled it with a fair amount of prompting rather than independently. That is the honest report, and it is fair: this is a genuinely sophisticated argument, and the goal here is that students follow it and feel its inevitability, not that every group generates it cold.
What I would want a teacher to know going in is that this lesson lands emotionally if you let it. The feeling that trigonometry and circle geometry were secretly the same subject is rare in a math class. It is worth protecting a quiet minute in consolidation for students to sit with that, rather than rushing on to the CYU. In fact, triangle “trigonometry” is a relatively recent it started as what James Tanton calls “circle-ometry”.