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Lesson Plan

Prove the Pythagorean identity sin^2(x) + cos^2(x) = 1 by analyzing similar right triangles

teaches Common Core State Standards CCSS.Math.Content.HSF-TF.C.8
teaches Common Core State Standards CCSS.Math.Content.HSG-SRT.C.8
teaches Common Core State Standards CCSS.Math.Practice.MP8
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Big Ideas: The Pythagorean identity, sin^2(x) + cos^2(x)=1, can be proved through analysis of a right triangle constructed inside the unit circle, and it can be used to solve problems involving unknown trigonometric function values. This task takes a geometric approach to deriving the Pythagorean identity by dilating an arbitrary right triangle so that it has a hypotenuse of 1 and then applying the Pythagorean Theorem. Students will see the two triangles are similar, and after substituting equivalencies, the Pythagorean identity is the result. The task debrief concludes by relating the Pythagorean identity to the unit circle from which it is derived. This lesson should come toward the end of a student's Algebra 2 study in Trigonometry or serve as a primer for more complex identities in precalculus. Vocabulary: reference triangle, similar triangles, Pythagorean Theorem, dilation Special Materials: No special materials are required for this task.
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