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| Credit- Degree applicable | | Effective Quarter: Fall 2020 | I. Catalog Information
| MATH 42 | Precalculus II: Trigonometric Functions | 5 Unit(s) |
| | Formerly: (See general education pages for the requirement this course meets.) Requisites: (Not open to students with credit in MATH 42H.)
Prerequisite: MATH 41 or MATH 41H (with a grade of C or better); or a satisfactory score on the College Level Math Placement Test within the last calendar year.
Advisory: EWRT 211 and READ 211, or ESL 272 and 273. Hours: Lec Hrs: 60.00
Out of Class Hrs: 120.00
Total Student Learning Hrs: 180.00 Description: The theory of trigonometric functions and their applications. |
| Student Learning Outcome Statements (SLO)
| | | • Student Learning Outcome: Formulate, construct, and evaluate trigonometric models to analyze periodic phenomena, identities, and geometric applications. |
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II. Course Objectives | A. | Define and evaluate trigonometric functions using both degree and radian measure |
| B. | Solve oblique and right triangles |
| C. | Solve arc length and sector area problems |
| D. | Graph and analyze the six trigonometric functions |
| E. | Apply trigonometric identities to simplify and evaluate trigonometric expressions and verify other identities |
| F. | Analyze the inverse trigonometric functions |
| G. | Solve trigonometric equations |
| H. | Define the polar coordinate system and introduce polar graphs |
| I. | Examine complex numbers in the complex plane |
| J. | Perform operations with 2D vectors |
| K. | Examine the logic of conditional and bi-conditional statements as they appear in mathematical statements |
III. Essential Student Materials | | Graphing calculator and/or computer software |
IV. Essential College Facilities V. Expanded Description: Content and Form | A. | Define and evaluate trigonometric functions using both degree and radian measure
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| 1. | Study angles, converting between radian and degree measures
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| 2. | Use the unit circle and right triangles to define the six trigonometric functions
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| 3. | Define and study properties of circular functions
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| 4. | Use special triangles to evaluate trigonometric function values of common angles in both degrees and radians |
| 5. | Use technology to evaluate trigonometric functions for any angle measured in radians or degrees |
| 6. | Investigate applications such as, but not limited to
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| a. | Finding the angle formed by two hands of a clock |
| b. | Reflection and refraction of light |
| c. | Historical contributions in the development of the trigonometric functions by various cultures such as the Middle East, Europe, and India
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| B. | Solve oblique and right triangles
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| 1. | Solve right triangles using trigonometric functions
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| a. | Solve for missing angles |
| b. | Solve for missing sides |
| 2. | Apply the law of sines and the law of cosines to oblique triangles |
| a. | Solve for missing angles |
| b. | Solve for missing sides |
| 3. | Investigate applications such as, but not limited to finding distances in problems arising in various applications from geometry, surveying, astronomy, physics, geography, navigation, and engineering |
| C. | Solve arc length and sector area problems
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| 2. | Solve circular motion problems for angular and linear velocity
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| 3. | Solve sector area problems
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| 4. | Investigate applications such as, but not limited to
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| a. | Motion of objects, such as pulleys and gears
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| D. | Graph and analyze the six trigonometric functions |
| 1. | Explore amplitude numerically, graphically, and symbolically |
| 2. | Explore period numerically, graphically, and symbolically
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| 3. | Explore vertical shifts and horizontal (phase) shifts numerically, graphically, and symbolically |
| 4. | Develop the general form of trigonometric functions |
| 5. | Investigate applications such as, but not limited to |
| a. | Sound waves and the relationship of frequency to period |
| b. | Using periodic functions to model periodic phenomena such as temperature, electric current, simple harmonic motion (of a spring, for example), light and electromagnetic waves, water waves, pressure of a plucked string |
| c. | Combining other functions with periodic functions to model phenomena such as damped harmonic motion |
| d. | Combining basic periodic functions to create other periodic functions; for example, Fourier series |
| E. | Apply trigonometric identities to simplify and evaluate trigonometric expressions and verify other identities |
| 1. | Develop and use basic identities
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| e. | Odd and even identities (negatives of angles) |
| 2. | Develop and use other trigonometric identities |
| a. | Sum and difference of two angles |
| d. | Product to Sum (optional) |
| 3. | Simplify trigonometric expressions |
| 4. | Verify trigonometric identities |
| 5. | Find exact trigonometric function values of special angles |
| F. | Analyze the inverse trigonometric functions |
| 1. | Define, evaluate, and graph the inverse trigonometric functions for sine, cosine, and tangent
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| 2. | Define, evaluate, and graph the inverse trigonometric functions for cotangent, secant and cosecant (optional)
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| 3. | Determine domain and range
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| 4. | Compose trigonometric and inverse trigonometric functions |
| G. | Solve trigonometric equations
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| 1. | Solve linear trigonometric equations |
| 2. | Solve quadratic trigonometric equations
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| 3. | Solve trigonometric equations involving multiples of angles
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| 4. | Investigate applications such as, but not limited to, solving equations that arise from modeling periodic phenomena such as temperature, electric current, simple harmonic motion, light and electromagnetic waves, water waves, pressure of a plucked string |
| H. | Define the polar coordinate system and introduce polar graphs
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| 1. | Plot points in the polar coordinate system |
| 2. | Convert between polar and rectangular coordinates
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| 3. | Convert between polar and rectangular equations |
| I. | Examine complex numbers in the complex plane
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| 1. | Perform operations on complex numbers
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| 3. | Write complex numbers in trigonometric/polar form |
| 4. | Use DeMoivre's Theorem to find powers and roots |
| J. | Perform operations with 2D vectors |
| 1. | Study two-dimensional vectors geometrically |
| a. | Define and use vectors as directed line segments |
| b. | Compute the magnitude and direction of vectors |
| c. | Define and use standard unit vectors |
| d. | Represent displacement, velocity, etc. as vector quantities |
| 2. | Study vector operations geometrically |
| a. | Define and use vector scalar multiplication |
| b. | Define and use vector addition |
| c. | Combine vectors geometrically using |
| 2. | The parallelogram method |
| 3. | Study two-dimensional vectors algebraically |
| a. | Express vectors algebraically as the sum of scalar multiples of the standard unit vectors |
| b. | Express vectors using ordered pairs |
| c. | Identify horizontal and vertical vector and scalar components of a vector |
| d. | Add vectors using scalar components and compute resultant vectors |
| e. | Study properties of vector addition and scalar multiplication |
| 4. | Study and use the dot product of two vectors |
| a. | Find the dot product of vectors |
| 1. | Find the magnitude of a vector |
| 2. | Find the angle between two vectors |
| 3. | Determine whether two vectors are orthogonal |
| c. | Study the properties of the dot product |
| d. | Find vector projections and resolve vectors into parallel and perpendicular vector components |
| 5. | Investigate application problems such as, but not limited to |
| a. | Static equilibrium problems |
| b. | Motion problems, such as sliding masses |
| d. | Historical development and use of vector quantities in solving problems in the sciences |
| K. | Examine the logic of conditional and bi-conditional statements as they appear in mathematical statements
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| 1. | Explore the relationships between a conditional statement and its converse, inverse, and contrapositive |
| 2. | Explore the use of conditional and bi-conditional statements in mathematical statements, definitions, and theorems |
VI. Assignments | A. | Required readings from text |
| B. | Problem-solving exercises, some including technology |
| D. | Optional project synthesizing various concepts and skills from the course content |
VII. Methods of Instruction | | Lecture and visual aids
Discussion of assigned reading
Discussion and problem-solving performed in class
In-class exploration of internet sites
Quiz and examination review performed in class
Homework and extended projects
Guest speakers
Collaborative learning and small group exercises
Collaborative projects
Problem solving and exploration activities using applications software
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VIII. Methods of Evaluating Objectives | A. | Periodic quizzes and/or assignments from sources related to the topics listed in the curriculum are evaluated for completion and accuracy in order to assess students' comprehension and ability to communicate orally and in writing of course content and provide timely feedback to students on their progress. |
| B. | Projects (optional)
Projects may be used to enhance the students' understanding of topics studied in the course in group or individual formats where communicating their understanding orally through classroom presentation or in writing. The evaluation to be based on completion and comprehension of course content and the students shall receive timely feedback on their progress. |
| C. | At least three one-hour exams without projects, or at least two one-hour exams with projects are required. In these evaluations the students are expected to provide complete and accurate solutions to problems that include both theory and applications by integrating methods and techniques studied in the course. Students shall receive timely feedback on their progress. |
| D. | One two-hour comprehensive final examination in which students are expected to display comprehension of course content and be able to choose methods and techniques appropriate to the various types of problems that cover course content. Students shall have access to the final exam for review with the instructor for a period determined by college and departmental rules. |
IX. Texts and Supporting References | A. | Examples of Primary Texts and References |
| 1. | Larson. Precalculus with Limits, 3rd Edition. Cengage, 2014 |
| 2. | Barnett, Ziegler, Byleen and Sobecki. Analytic Trigonometry with Applications, 10th Edition. Wiley, 2008
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| 3. | Lial, Hornsby, Schneider and Daniels. Trigonometry, 10th Edition. Pearson, 2012 |
| B. | Examples of Supporting Texts and References |
| 1. | Blatner, David. The Joy of Pi. Walker and Co., 1997 |
| 2. | Math Department Activity and Multicultural Resource Binder, available in the PSME Division Office |
| 3. | Joseph, George Gheverghese. The Crest of the Peacock: Non-European Roots of Mathematics, 3rd Edition. Penguin Books, 2010 |
| 4. | Heilbron, J. L. Geometry Civilized: History, Culture and Technique. Clarendon Press, 1998
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| 5. | Maor, Eli. Trigonometric Delights. Princeton University Press, 1998 |
| 6. | Nahin, Paul. An Imaginary Tale: The Story of Sqrt(-1). Princeton University Press, 1998 |
| 7. | Historical Topics for the Mathematics Classroom. National Council of Teachers of Mathematics, Inc., 1998 |
| 8. | Nelson, David, George Gheverghese Joseph and Julian Williams. Multicultural Mathematics: Teaching Mathematics from a Global Perspective. Oxford University Press, 1993 |
| 9. | Rieder, John and Larry Smith, editors. Multiculturalism and Representation: Selected Essays. East-West Center, 2001 |
| 10. | Alcoze, Thom and Miriam Barrios-Chacon. Multiculturalism in Mathematics, Science and Technology: Readings and Activities. Clarendon Press, 1999
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| 11. | The MacTutor History of Mathematics Archive. School of Mathematics and Statistics, University of St. Andrews, Fife, Scotland. http://www-groups.dcs.st-and.ac.uk/~history/Indexes/historyTopics.html, http://www-groups.dcs.st-and.ac.uk/~history |
| 12. | Smith, Karl. Trigonometry, 4th Edition. Thomson Brooks/Cole, 2005 |
| 13. | Connally, Hughes-Hallett, Gleason, et al. Functions Modeling Change, 4th Edition. Wiley, 2011
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| 14. | Sullivan, M. Trigonometry, a Unit Circle Approach, 7th Edition. Prentice-Hall, 2005 |
| 15. | Aratari. Trigonometry, a Circular Function Approach. Addison-Wesley, 2004 |
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