In a previous lesson, students built a right triangle table. Each group had a pair of complementary angles, so they began to make conjectures about cosine and sine of complementary angles before they learned the terms cosine or sine. In this lesson, students do some calculations to remind them of their previous conjectures and then prove \(\sin(\theta)=\cos(90\theta)\).
Throughout this lesson there is a focus on precision of language. The warmup prompts students to compare four triangles. The Which One Doesn't Belong? routine gives students a reason to use language precisely (MP6). The following activity asks students to explain how they got the same answers as their partner despite being assigned different triangles (the pairs of triangles were congruent but had different angles provided). In the final activity students write a draft of a proof, work with their group to refine the group proof, and then have a whole class discussion on how to clearly communicate ideas using words and diagrams.
Lesson overview
 8.1 Warmup: Which One Doesn’t Belong: Four Triangles (5 minutes)
 8.2 Activity: Twin Triangles (15 minutes)

8.3 Activity: Explain the Connection (15 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 8.4 Cooldown: Cosine’s Complement (5 minutes)
Learning goals:
 Explain the relationship between the cosine and sine of complementary angles (using words and other representations).
Learning goals (student facing):
 Let’s connect cosine and sine.
Learning targets (student facing):
 I can explain why \(\sin(\theta)=\cos(90\theta)\).
Required materials:
 Scientific calculators
 Tools for creating a visual display
Standards:
 This lesson builds on the standard: CCSS.HSGSRT.C.6MS.GSRT.6MO.G.SRT.C.5
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