Overview. Essential Questions. Precalculus, Quarter 2, Unit 2.4 Interpret, Solve, and Graph Inverse Trigonometric Functions

Transcription

1 Trigonometric Functions Overview Number of instruction days: 3 5 (1 day = 53 minutes) Content to Be Learned Use restricted domains in order to construct inverse Use inverse trigonometric functions to solve equations found in modeling contexts and interpret the solutions. Graph the inverses of sine, cosine, and tangent functions. Solve real-word situations using inverse Mathematical Practices to Be Integrated 4 Model with mathematics. Use inverse trigonometric functions to model real-world situations. 5 Use appropriate tools strategically. Use graphing calculators to graph inverse 6 Attend to precision. Make explicit use of definitions by using the appropriate domain and range for inverse Essential Questions What is true about all inverse functions? Why is the domain restricted when graphing inverse trigonometric functions? What are the similarities and differences between an inverse trigonometric function and the inverse of previously learned functions? Where are inverse trigonometric functions used in real-world situations? Providence Public Schools D-51

2 Version 4 Trigonometric Functions (3 5 days) Standards Common Core State Standards for Mathematical Content Functions Building Functions F-BF Build new functions from existing functions F-BF.4 Find inverse functions. d. (+) Produce an invertible function from a non-invertible function by restricting the domain. Trigonometric Functions F-TF.7 (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. ACT s College Readiness Standards: Mathematics F.3.k Identify and graph inverse sine, cosine, and tangent functions Common Core State Standards for Mathematical Practice 4 Model with mathematics. F-TF Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that D-52 Providence Public Schools

3 Precalculus, Quarter 2, Unit 2.4 Trigonometric Functions (3 5 days) Version 4 technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 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 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 Students have graphed functions in Algebra I, Geometry, and Algebra II. They have found domain, range, x- and y-intercepts, maximum and minimum, slope, inverses, and transformations of functions. In Geometry, students graphed circles and found length of the arc. In Algebra II, students studied radian measure as the length of an arc on the unit circle. They also used the unit circle on a coordinate plane to find the trigonometric functions for real numbers and interpreted radian measures, and they modeled Current Learning This unit reinforces trigonometric functions and introduces the inverses of Students use the inverse trigonometric functions to solve problems in real-world situations. Students will use the appropriate domain to graph inverse Future Learning In AP Calculus AB, students will discover that the arctangent function is the antiderivative of 1 dx. 1 x 2 Additional Findings There are no additional findings for this unit. Providence Public Schools D-53

4 Version 4 Trigonometric Functions (3 5 days) Assessment When constructing an end-of-unit 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. Standards-based 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 performance-based tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. 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. Find the inverse of a function that is not one-to-one by restricting the domain. Use the inverse of trigonometric functions to solve real-world equations. Identify and graph inverse Use technology to evaluate the solutions to the inverse trigonometric functions, and interpret their meaning in terms of context. Learning Objectives Instruction Students will be able to: Graph inverse sine, cosine, and tangent functions by hand and with technology. Find principal values of inverse Solve real-world situations using inverse Demonstrate understanding of concepts and skills related to inverse Resources Advanced Mathematical Concepts: Precalculus with Applications, Glencoe, 2006, Teacher Edition and Student Edition Section 6-8 (pp ) TeacherWorks All-In-One Planner and Resource Center CD-ROM Exam View Assessment Suite Glencoe McGraw- Hill Online 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. D-54 Providence Public Schools

5 Precalculus, Quarter 2, Unit 2.4 Trigonometric Functions (3 5 days) Version 4 Materials TI-Nspire graphing calculators, chart paper Instructional Considerations Key Vocabulary arcsin, Arcsin arccos, Arccos arctan, Arctan arccsc noninvertible arccot arcsec principal values invertible Planning for Effective Instructional Design and Delivery Reinforced vocabulary from previous grades or units: cos, inverse function, sin, and tan. Have students work in pairs to make pictographic representations (nonlinguistic representations) to deepen their knowledge of inverse Assign a portion of the class to work on each of the inverse trigonometric functions (arcsin, arcos, arctan, arccot, arcsec, arccsc). Have each group of students graph on a large piece of chart paper the parent function. On the same graph, students should graph the inverse function, for which they may use tables of values (they may use the ones they have from the parent functions as a starting point). Also have the groups write a scenario that would fit the inverse function they have been assigned, have all students do a gallery walk to see the graphs, and then have groups share out their scenarios with the whole group. As an extension, chart what students notice and the similarities and the differences between the trigonometric inverse functions and other inverse functions. To ascertain the level of student understanding, use correct student-created scenarios as questions on the assessment. As a modeling tool, all students should use TI-Nspire calculators to graph each trigonometric function and its inverse to help them understand why the domain on the inverse function must be restricted (in order for it to be a function). Students should also use the calculators to evaluate the accuracy of the solutions to the equations they have solved, including the real-world problems. In Section 6-8, when using inverse functions to solve trigonometric equations (Example 3), make sure to model for students how to check their solutions using a graphing calculator. Additional examples can be found for Section 6-8 on the TeacherWorks All-In-One Planner and Resource Center or on the Glencoe website provided in the resource section. Providence Public Schools D-55

6 Version 4 Trigonometric Functions (3 5 days) Notes D-56 Providence Public Schools