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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 12 Precalculus Module 4 Trigonometry

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Participant Poll Classroom teacher Math trainer or coach Principal or school leader District representative / leader Other

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Session Objectives Participants will be able to derive trigonometric identities from the addition and subtraction formulas. Participants will understand restricting the domain of trigonometric functions to find inverse trigonometric functions. Participants will enrich their knowledge and experience in order to implement Module 4 with confidence and success.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Agenda Today Module 4, Topic A – Trigonometric Functions Module 4, Topic B – Trigonometry and Triangles Module 4, Topic C – Inverse Trigonometric Functions Discussion for Implementation

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Module 4: Trigonometry Module Overview 3 Topics 14 Lessons 20 days Topic A – Trigonometric Functions Mid-Module Assessment (After Topic A) Topic B – Trigonometry and Triangles Topic C – Inverse Trigonometric Functions End of Module Assessment (After Topic C)

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Module 4: Trigonometry Standards Addressed in this presentation: (+) G-SRT.D.9:Derive formula A = ½ ab sin(C) for the area of a triangle (+) G-SRT.D.10: Prove the Law of Sines and Law of Cosines (+) G-SRT.D.11:Understand and apply the Law of Sines & Law of Cosines (+) F-TF.B.6:Restrict the domain of a trigonometric function to find its inverse (+) F-TF.B.7:Use inverse functions to solve trigonometric equations (+) F-TF.C.9:Prove addition & subtraction formulas for sine & cosine Others in the module: G-C.A.4, F-TF.A.3, F-TF.A.4

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Module 4: Trigonometry Topic A Finding values of trigonometric functions using properties of special triangles and the unit circle. Establishing and using the addition and subtraction formulas for sine and cosine. Tangent lines and the tangent function Topic B The area formula The Law of Sines and Law of Cosines Topic C Inverse trigonometric functions

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A: Trigonometric Functions Topic Overview 6 lessons (1-6). Find the values of the trigonometric functions at multiples of /6, /4, /3 using special triangles. Use the unit circle to express values of the trigonometric functions for - x, + x, 2 - x in terms of their values for x Use the unit circle to explain symmetry and periodicity of trigonometric functions. Prove and use the addition and subtraction formulas for sine, cosine, and tangent Construct the tangent lines to a circle from an exterior point.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 3: Addition and Subtraction Formulas Suppose that a rider on the outer edge of a carousel rotates /4 radians from the starting position at (1,0) and then stops. How can we determine the rider’s position? What if the rider had rotated /3 radians? What if the rider had rotated 7 /12 radians? Can we just sum the front/back and right/left positions from /3 and /4 to find her new location?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 3: Addition and Subtraction Formulas We develop the subtraction formula for sine by working through Example 1. Take a look at that Example now.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 3: Addition and Subtraction Formulas

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 3: Addition and Subtraction Formulas Work through Exercises 3-5. These exercises verify the difference formula for tangent, evaluate sine and cosine at “non-special” values, and verify some other identities.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 4: Addition and Subtraction Formulas Work through Exercise 1, 2 and 3 in Lesson 4.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 4: Addition and Subtraction Formulas

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Summing Up Topic A Let’s Review What major concepts and/or themes did you notice in Topic A? How do you think those topics and themes will apply as we move into studying the Law of Sines, Law of Cosines, and inverse trigonometric functions?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic B: Trigonometry and Triangles Topic Overview 4 lessons (7-10). An area formula for triangles. The Law of Sines and the Law of Cosines. Applying the Law of Sines and the Law of Cosines.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 7: An Area Formula for Triangles Lesson 7 is devoted to developing and using the formula for a triangle with sides of length b and c and included angle A. This is directly specified by standard G-SRT.C.9: (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 7: An Area Formula for Triangles Work through Problems 1 – 6 in the Exploratory Challenge in this lesson to derive the area formula.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 7: An Area Formula for Triangles Discussion questions: How are all of these problems similar? What did you have to do differently in Exercise 4 to determine the height of the triangles?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 7: An Area Formula for Triangles Work through Exercises 7 and 8 to establish the formula.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 7: An Area Formula for Triangles Exercise 8 established the area formula for a triangle with sides of length b and c and included angle A. Complete Exercise 9 & 10 to put this formula to use.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 7: An Area Formula for Triangles Discussion question: What is the importance of Exercise 10f?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 8 & 9: The Law of Sines and the Law of Cosines

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 10: Putting the Law of Cosines and Law of Sines to Use Begin with the Opening Exercise in Lesson 10.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 10: Putting the Law of Cosines and Law of Sines to Use Continue with Exercises 3-12 in Lesson 10.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Summing Up Topic B Let’s Review What major concepts and/or themes did you notice in Topic B? Have have the topics and themes of Topics A & B prepared you (and students) to understand the upcoming work in Topic C on Inverse Trigonometric Functions?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic C: Inverse Trigonometric Functions Topic Overview 4 lessons (11-14). Revisiting the graphs of the trigonometric functions. Defining the inverse trigonometric functions. Modeling with inverse trigonometric functions

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 12: Inverse Trigonometric Functions Question: Are any of the trigonometric functions invertible?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 12: Inverse Trigonometric Functions Work through the Opening Exercises in Lesson 12.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 12: Inverse Trigonometric Functions We start with our original equation: Then, we interchange x and y: Then we solve for y. But, we can’t solve for y. What do we do?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 12: Inverse Trigonometric Functions The inverse of the sine function is denoted by What is the domain of the inverse sine function? What is the range of the inverse sine function?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 12: Inverse Trigonometric Functions What is the value of ? What is the value of ? Why?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 12: Inverse Trigonometric Functions Work through Exercises 1-3 in Lesson 12.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 12: Inverse Trigonometric Functions Question to think about: What is the difference between solving the equation and evaluating the expression ?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lessons 13-14: Modeling with Inverse Trigonometric Functions In Lessons 13 and 14, students model real-life situations using inverse trigonometric functions.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lessons 13-14: Modeling with Inverse Trigonometric Functions Work through the Statue of Liberty Example in Lesson 13.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lessons 13-14: Modeling with Inverse Trigonometric Functions Work through Exercise 3 on interrelated rabbit and coyote populations in Lesson 14.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Summing Up Topic C Let’s Review What major concepts and/or themes did you notice in Topic C? How do those topics and themes rely on topics studied earlier in the module?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Themes of Module 4: Trigonometry Trigonometric identities and properties of trigonometric functions The Law of Sines and Law of Cosines Inverse trigonometric functions.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Biggest Takeaway What is your biggest takeaway from this study of Module 4? Given your role, how can you support successful implementation at your school?

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Section 12-4. Recall Then you applied these to several oblique triangles and developed the law of sines and the law of cosines.

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