The Law of Cosines

Day 1 begins with one sentence of framing: ‘All year we have seen that creating hidden triangles, especially right triangles, unlocks problems that seemed impossible. Keep that in mind, even if you already have a visible triangle.’ Then the task: find a relationship between the three sides of any triangle, even one without a right angle.

 

All groups drew altitudes — the right move. The main challenge was managing variables. Naming the two segments of the base x and (b−x), where b is already a variable, was the key move. Once introduced, most groups set up the two Pythagorean equations and recognized that substitution for h was the path forward.

 

The algebra errors were consistent and predictable. (b−x)² written as b²−x² appeared in multiple groups across both classes. The fix is the same numerical analogy as in Lesson 7.4: ‘What is (10−3)²? Is it 100−9?’ Students self-correct immediately. The order-of-operations error — treating 2bc·cos A as something to be subtracted entirely before multiplying by cosine — persisted even after multiple whole-class corrections. Write an equivalent expression without trig: ‘How would you evaluate 240−236(3)?’ They get it right. Then: ‘240−236·cos(15°) works the same way.’ This eventually lands.

 

Groups that did not fully derive the Law of Cosines still did valuable work. Many wrote h = c·sin(A) — which is essentially a key derivation step on its own. Mathematics does not have a clock. The work being done was real. Name this explicitly when groups feel they failed because they didn’t finish: what you wrote was genuine mathematics, it was going somewhere true, and the fact that class ended before you reached the destination is not a failure.

 

Day 2 opens with the Pythagorean theorem as a special case — substituting C = 90° into the Law of Cosines makes cos(90°) = 0 and the correction term vanishes. Then comes the ambiguous case problem. Post: ‘In a triangle, A = 30°, a = 6, and b = 10 (b adjacent to A). Find the triangle.’

 

The 30° angle will tempt many groups toward 30-60-90 reasoning. Let them pursue it. Then ask: ‘Are you assuming anything? If so, is that assumption justified?’ Groups that assumed 30-60-90 eventually disproved it using the Law of Sines on their results. Two groups prefaced their work with ‘well, assuming…’ — which is exactly the right epistemic language. Name it: they have identified that they are introducing a premise without justification. That is not the same as having a flaw — but the premise must be tested.

 

The problem requires Law of Sines, Law of Cosines, triangle angle sum theorem, and linear pair theorem. At the close, point this out explicitly.