Transcript Precalculus and Advanced Topics Module 4

NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
A Story of Functions
Trigonometry
NYS COMMON CORE MATHEMATICS CURRICULUM
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Session Objectives
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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.
NYS COMMON CORE MATHEMATICS CURRICULUM
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
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
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)
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Module 4: Trigonometry
• (+) 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
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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A Story of Functions
Topic A: Trigonometric Functions
Topic Overview
• 6 lessons (1-6).
• Find the values of the trigonometric functions at multiples of p/6, p/4,
p/3 using special triangles.
• Use the unit circle to express values of the trigonometric functions for
p - x, p + x, 2p - 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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A Story of Functions
Formulas
• Suppose that a rider on the outer edge of a carousel
rotates p/4 radians from the starting position at (1,0)
and then stops. How can we determine the rider’s
position?
just sum the front/back and right/left positions from
p/3 and p/4 to find her new location?
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A Story of Functions
Formulas
We develop the subtraction formula for sine by
working through Example 1.
Take a look at that Example now.
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Formulas
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
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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A Story of Functions
Formulas
Work through Exercise 1, 2 and 3 in
Lesson 4.
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A Story of Functions
Formulas
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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?
A Story of Functions
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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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.
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.
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?
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.
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.
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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A Story of Functions
Lesson 8 & 9: The Law of Sines and the Law
of Cosines
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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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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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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?
A Story of Functions
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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A Story of Functions
Lesson 12: Inverse Trigonometric Functions
Question: Are any of the trigonometric functions
invertible?
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A Story of Functions
Lesson 12: Inverse Trigonometric Functions
Work through the Opening Exercises in Lesson 12.
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 12: Inverse Trigonometric Functions
• Then, we interchange x and y:
• Then we solve for y.
• But, we can’t solve for y. What do we do?
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?
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 12: Inverse Trigonometric Functions
• What is the value of
?
• What is the value of
?
• What is the value of
? Why?
• What is the value of
? Why?
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 12: Inverse Trigonometric Functions
Work through Exercises 1-3 in Lesson 12.
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 12: Inverse Trigonometric Functions
What is the difference between solving the equation
and evaluating the expression
?
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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.
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.
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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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?
A Story of Functions
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.