Lesson 7.5
The Shape of Sine
Focus: graphical representation — four representations connected — the Desmos moment — optional lesson
This lesson is optional. See the note in the unit introduction for how to think about the time decision. The full lesson materials are in the downloadable documents.
The lesson asks groups to plot y = sinθ and y = cosθ on the same axes from 0° to 90° using exact values they can justify. The goal is to connect all four representations of the function — verbal (ratio definition), symbolic (sinθ = opp/hyp), numerical (the class table), and graphical (the curve they plot). Each representation reveals something the others hide. The graph shows how the ratios change. The table shows specific values. The formula connects to geometry.
The consolidation closes with the Desmos animation showing a point moving around the unit circle, tracing sine and cosine waves. Show one full cycle without narration. Ask where sine grows fastest. Then stop: ‘These waves have a name. You will study them formally in precalculus. What questions does this raise for you?’ Leave students with a question they cannot yet answer. The feeling that what they have learned opens into something larger is exactly right.