The “Why” of Trigonometry (CoRD)
Michael Fulkerson, University of Central Oklahoma
Kristi Karber, University of Central Oklahoma
Scott Williams, University of Central Oklahoma
Trigonometry is of primary importance in any precalculus course. But often students learn the basics of trigonometric functions and identities without really understanding “why” everything works and how everything “fits together.” A frequently heard complaint is that there is too much memorization: “How can I remember all these identities?” This CoRD seeks to address these issues, while developing the targeted topics listed above.
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Why is the Pythagorean Theorem true?
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This lesson provides students a hand-on proof of why the Pythagorean Theorem is true. Students will use manipulatives to create a geometric shape whose area, when calculated using differing methods, establishes the theorem.
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Special Triangles: 45-45-90, 30-60-90, and 15-75-90
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In this lesson, students will investigate the properties of three special right triangles: the 45-45-90 triangle, the 30-60-90 triangle, and the 15-75-90 triangle. The goal is to explore and derive the exact proportions of the side lengths of these triangles through student-centered inquiry. Students will begin by making conjectures about the side lengths based on geometric reasoning and estimation, and then verify their conjectures through various diagrams involving the triangles and then using the Pythagorean Theorem. Students also solve some application problems related to the three special triangles.
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The Angle Sum Identities for Cosine and Sine
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The goal of this activity is for students to derive the angle sum identity for cosine. Students will use GeoGebra (for a geometric visualization), the distance formula, and algebra to understand and prove the identity.
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Avoid Memorization: Deriving Identities
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In this activity, students will use their critical thinking and algebraic skills to derive multiple trigonometric identities. They will understand how a few trigonometric identities and facts can be used to create many other identities, thereby reducing their need for memorization. Problem-solving skills are encouraged as students select an appropriate substitution for a variable in an equation or add a clever “0.” Thus, this activity encourages students to be active participants in their own education as they deepen their understanding of trigonometry and develop a broader appreciation for the cohesive nature of mathematical principles and their applications.
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This work is licensed under CC BY-NC-SA 4.0
