Lesson 7.6
The Law of Sines
Focus: area from two sides and included angle — Law of Sines derived — Mount Everest problem
Day 1 of this lesson was one of the best days of group work in the entire year. The task is simple: each group receives a fully labeled triangle card and is asked to find its area. Then find it again a different way. Then a third way. Every group used trig to write the altitude on their own, without prompting. That is worth naming when it happens — students are applying tools they understand, not following a procedure.
The most common mistake: confusing the altitude dropped with the perpendicular bisector of the base. Students assumed it bisected the base. This led to a productive conversation: the antecedent of a theorem must be met before we can accept the consequent. The altitude is only the perpendicular bisector if the triangle is isosceles. The theorem from Unit 4 requires the isosceles condition. This is the logical thread of the course surfacing in a trigonometry context.
When groups have found the area three ways, ask them to write all three expressions in general form using letters a, b, c, A, B, C. The algebraic derivation of the Law of Sines from those three equal expressions follows naturally: if ½bcsin A = ½acsin B = ½absin C, divide by ½abc, and the Law of Sines emerges. Name the inductive/deductive distinction at this moment: every group found the same structure in their own specific triangle — that was inductive evidence. The algebraic derivation using a general triangle is deductive proof.
Day 2: the Mount Everest problem replaced the original outdoor goal-post measurement task and was a significant improvement. The setup: why can’t we use any indirect measurement technique from Unit 6 to measure Everest? The answer: we cannot get the horizontal distance to the base — the terrain is the Himalayas. Two surveyors at different positions on flat land measure elevation angles to the summit. The baseline between them is known. The Law of Sines gives the slant distance; right-angle trig gives the height.
Some groups found the hypotenuse of the inner right triangle; some found it of the outer. Both approaches work and led to rich conversation among groups about why. Our answer was approximately 29,728 ft vs. the actual 29,032 ft — a 2.4% error from using whole-degree angles rather than degrees-minutes-seconds. This is the right moment to mention that degrees break into minutes and seconds, and that surveyors measure to arc-seconds. Our error came from measurement precision, not from mathematical error.