In this lesson, students prove the identity . Students begin by doing some calculations to collect data. Using their observations, they build conjectures about cosine and sine of complementary angles. After reaching a consensus, they then prove the identity .

Throughout this lesson there is a focus on precision of language. The Warm-up prompts students to compare four triangles. The Which Three Go Together? 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 next activity students write a draft of a proof, work with their group to refine the group’s proof, and then have a whole-class discussion on how to clearly communicate ideas using words and diagrams.

This lesson includes an optional activity that allows students to apply the identity and generate more examples to further explore the relationship between cosine and sine. In the activity, students determine which angles cause sine to be less than cosine, equal to cosine, and greater than cosine. The activity is optional because students practice an application that is beyond grade level.