Overview. Essential Questions. Precalculus, Quarter 2, Unit 2.5 Proving Trigonometric Identities. Number of instruction days: 5 7 (1 day = 53 minutes)


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1 Precalculus, Quarter, Unit.5 Proving Trigonometric Identities Overview Number of instruction days: 5 7 (1 day = 53 minutes) Content to Be Learned Verify proofs of Pythagorean identities. Apply Pythagorean, reciprocal, quotient, symmetry, and opposite angle identities to prove other identities. Prove sum and difference identities for sine, cosine, and tangent. Mathematical Practices to Be Integrated Reason abstractly and quantitatively. Make sense of problems and persevere in proving trigonometric identities. 3 Construct viable arguments and critique the reasoning of others. Build a logical progression of statements in order to confirm trigonometric identities. 6 Attend to precision. Examine claims and make explicit use of definitions in verifying trigonometric identities. Essential Questions What is the relationship between the Pythagorean Theorem and the Pythagorean identities in trigonometry? Why is the study of trigonometric identities important? In what ways can trigonometric identities be used to simplify trigonometric expressions? Providence Public Schools D57
2 Precalculus, Quarter, Unit.5 Proving Trigonometric Identities (5 7 days) Standards Common Core State Standards for Mathematical Content Functions Trigonometric Functions FTF Prove and apply trigonometric identities FTF.8 Prove the Pythagorean identity sin cos 1 and use it to findsin, cos, or tan given sin, cos, or tan and the quadrant of the angle. FTF.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Common Core State Standards for Mathematical Practice Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols D58 Providence Public Schools
3 Proving Trigonometric Identities (5 7 days) Precalculus, Quarter, Unit.5 they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Clarifying the Standards Prior Learning In Algebra I and Geometry, students solved problems algebraically and graphically and solved problems involving systems of linear equations. Also in Algebra I, students used the Pythagorean Theorem and literal equations to solve problem situations. In Algebra II, students were introduced to factoring, completing the square, and solving and interpreting solutions involving polynomial, piecewise, absolute value, rational, and radical functions. Geometry students used proofs in their studies. Algebra II students proved the Pythagorean identity sin cos 1 and used it in their study of the unit circle. Current Learning Precalculus students use trigonometric identities other than the Pythagorean identity for the first time. Students use the basic trigonometric identities to verify other identities, and they develop and use the sum and difference identities. Future Learning Calculus students will use these identities throughout their work with differentiation and integration techniques. Additional Findings There are no additional findings for this unit. Assessment When constructing an endofunit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standardsbased mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performancebased tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. Providence Public Schools D59
4 Precalculus, Quarter, Unit.5 Proving Trigonometric Identities (5 7 days) The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Prove the Pythagorean identity sin (θ) + cos (θ) = 1. Find numerical values of trigonometric functions. Use the basic identities to verify other trigonometric identities. Apply trigonometric identities to model realworld context. Formulate trigonometric identity equations to solve realworld problems. Learning Objectives Students will be able to: Instruction Prove and use the Pythagorean identity sin (θ) + cos (θ) = 1. Identify and use reciprocal identities, quotient identities, Pythagorean identities, symmetry identities, and oppositeangle identities. Use the basic trigonometric identities to verify other identities. Use sum and difference identities for the sine, cosine, and tangent functions to solve real world problems. Demonstrate understanding of concepts and skills learned in this unit. Resources Advanced Mathematical Concepts: Precalculus with Applications, Glencoe, 006, Teacher Edition and Student Edition Section 71 (pp ) Section 7 (pp ) Section 73 (pp ) with supplemental material for modeling realworld situations TeacherWorks AllInOne Planner and Resource Center CDROM Exam View Assessment Suite Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations. Materials TINspire graphing calculators, formula chart (See the Planning for Effective Instructional Design and Delivery section. D60 Providence Public Schools
5 Proving Trigonometric Identities (5 7 days) Precalculus, Quarter, Unit.5 Instructional Considerations Key Vocabulary basic trigonometry identities quotient identities oppositeangle identities Pythagorean identities symmetry identities Planning for Effective Instructional Design and Delivery Reinforced vocabulary from previous grades or units: cosecant, cosine, cotangent, secant, sine, and tangent. Precalculus students are required by the Common Core State Standards to prove the addition and subtraction formulas for sine, cosine, and tangent. The extension of proofs of trigonometric identities is considered vital for the depth and rigor of study expected for Precalculus students. In helping students navigate the complexities of proving trigonometric identities and solving realworld problems, you can use nonlinguistic representations. A graphic representation in the form of a formula sheet will provide students with a visual and eventually a mental image to help them remember the basic formulas to use in proofs or in solving realworld problems. Students can also make their own formula sheet, leaving room to add completed examples of realworld problems that can be solved using that particular formula. By adding the completed examples, students will have access to the algebraic manipulations that must be accomplished to solve trigonometric equations. Proofs are equality relationships between two mathematical expressions; they are used to simplify algebraic expressions and to solve algebraic equations. No single method works for all identities. However, following certain steps might help. To verify an identity, students may start by using basic identities to transform the more complicated side of the identity into the same expression as the other side. Students may also use the technique of transforming the two sides of the identity into the same expression. To do this work, students do not have to have memorized all the identities; their learning could be supported by the use of a formula sheet with the principal identities. Have students verify trigonometric identities at the board or using a projection device, explaining their understanding to other students as they work through the process. Have students work in pairs or groups of three to verify trigonometric identities on large chart paper. Give each group a different identity, and be careful to have all different kinds of techniques represented in the examples that must be verified. Have students share their different ways of accomplishing this task. After each group has shared their verification, ask the whole group if they have another way to verify the same identity. Note: The same process could be used in having students solve realworld problems. Providence Public Schools D61
6 Precalculus, Quarter, Unit.5 Proving Trigonometric Identities (5 7 days) Notes D6 Providence Public Schools
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