Download 2014-15 Math Analysis/Trigonometry Course Purpose

2014-15 Math Analysis/Trigonometry
Course Purpose
The primary purpose of this course is to prepare the students for future studies in mathematics; in
particular, Calculus. The students will be introduced to the major concepts and tools for a
thorough understanding of the study of trigonometry, as well as the studies of conic sections and
logarithms, all with the primary goal of bolstering weaknesses commonly found in Calculus
students.
Course Outline
Unit One: Introduction to Trigonometry ( weeks)
1.1 Angle Measure and Special Triangles
A. The students will use the vocabulary associated with a study of angles and triangles.
B. The students will find and recognize coterminal angles.
C. The students will find fixed ratios of the sides of special triangles.
1.2 Properties of Triangles; Similar Triangles
A. The students will recognize and classify different types of triangles.
B. The students will analyze triangles using their common properties.
C. The students will solve applications using similar triangles.
1.3 Trigonometry: The Coordinate Plane
A. The students will define the six trigonometric functions using the coordinates of a point.
B. The students will analyze the signs of the trigonometric functions of an angle.
C. The students will evaluate the trigonometric functions of quadrantal angles.
1.4 Fundamental Identities and Families of Identities
A. The students will use fundamental identities to understand identity families.
B. The students will verify identities using the fundamental identities and algebra skills.
C. The students will express a given trig function in terms of the other five.
Unit Two: Right Triangles and Static Trigonometry ( weeks)
2.1 The Right Triangle View of Trigonometry
A. The students will find the values of six trigonometric functions from their ratios.
B. The students will bridge the two definitions of the trigonometric functions by positioning
a right triangle in the coordinate plane.
C. The students will use cofunctions and complements to write equivalent expressions.
2.2 Solving Right Triangles
A. The students will solve a right triangle given one angle and one side.
B. The students will solve a right triangle given two sides.
2.3 Applications of Static Trigonometry
A. The students will solve applications involving angles of elevation and depression.
B. The students will solve applications involving angles of rotation.
C. The students will solve general applications of right triangles.
2.4 Extending Beyond Acute Angles
A. The students will find the reference angle of any nonquadrantal angle.
B. The students will use reference angles to evaluate the trig functions for any
nonquadrantal angle.
C. The students will solve applications using the trig functions of any angle.
Unit Three: Radian Measure and Dynamic Trigonometry ( weeks)
3.1 Angle Measure in Radians
A. The students will use radians for angle measure.
B. The students will find the radian measure of the standard angles.
C. The students will convert between degrees and radians for nonstandard angles.
3.2 Arc Length, Velocity, and the Area of a Circular Sector
A. The students will use radians to compute the length of a subtended arc.
B. The students will solve applications involving angular velocity and linear velocity.
C. The students will calculate the area of a circular sector.
3.3 The Unit Circle
A. The students will locate points on a unit circle and use symmetry to locate other points.
B. The students will use special triangles to find points on a unit circle and locate other
points using symmetry.
C. The students will define the six trig functions in terms of a point on the unit circle.
3.4 The Trigonometry of Real Numbers
A. The students will define the six trig functions in terms of a real number t
B. The students will find the real number t corresponding to special values of sin t, cos t, and
tan t
C. The students will find the real number t corresponding to any trig function value
Unit Four: Trigonometric Graphs and Models ( weeks)
4.1 Graphs of Sine and Cosine Functions
A. The students will graph f(t) = sin t using special values and symmetry
B. The students will graph f(t) = cos t using special values and symmetry
C. The students will graph sine and cosine functions with various amplitudes and periods
D. The students will write the equation for a given graph
4.2 Graphs of Cosecant, Secant, Tangent, and Cotangent Functions
A. The students will graph the functions y = Acsc(Bt) and y = Asec(Bt)
B.
C.
D.
E.
The students will graph y = tan t using asymptotes, zeroes, and the ratio sin t/cos t
The students will graph y = cot t using asymptotes, zeroes, and the ratio cos t/sin t
The students will identify and discuss important characteristics of y = tan t and y = cot t
The students will graph y = Atan(Bt) and y = Acot(Bt) with various values of A and B
4.3 Transformations of Trigonometric Graphs
A. The students will graph vertical translations of y = Asin(Bt) and y = Acos(Bt)
B. The students will graph horizontal translations of y = Asin(Bt) and y = Acos(Bt)
C. The students will identify and apply important characteristics of y = Asin(Bt+or-C) + D
D. The students will apply vertical and horizontal translations to y = Acsc(Bt), y = Atan(Bt),
and Acot(Bt)
4.4 Trigonometric Applications and Models
A. The students will model simple harmonic motion
B. The students will solve applications using sinusoidal models
C. The students will solve applications involving the tangent, cotangent, secant, and
cosecant functions
Unit Five: Trigonometric Identities ( weeks)
5.1 More on Verifying Identities
A. The students will identify and use identities due to symmetry
B. The students will verify general identities
C. The students will use counterexamples and contradictions to show an equation is not an
identity
5.2 The Sum and Difference Identities
A. The students will develop and use sum and difference identities for cosine
B. The students will use the cofunction identities to develop the sum and difference
identities for sine and tangent
C. The students will use the sum and difference identities to verify other identities
5.3 The Double-Angle and Half-Angle Identities
A. The students will derive and use the double-angle identities for cosine, tangent, and sine
B. The students will develop and use the power reduction and half-angle identities
5.4 The Product-to-Sum and Sum-to-Product Identities
A. The students will derive and use the product-to-sum and sum-to-product identities
B. The students will solve applications using these and other identities
Unit Six: Inverse Functions and Trigonometric Equations ( weeks)
6.1 One-to-One and Inverse Functions
A. The students will identify one-to-one functions
B. The students will explore inverse functions using ordered pairs
C. The students will find inverse functions using an algebraic method
D. The students will graph a function and its inverse
E. The students will solve applications of inverse functions
6.2 The Inverse Trigonometric Functions and Their Applications
A. The students will find and graph the inverse sine function and evaluate related
expressions
B. The students will find and graph the inverse cosine function and tangent function and
evaluate related expressions
C. The students will apply the definition and notation of inverse trig functions to simplify
compositions
D. The students will find and graph inverse functions for sec x, csc x, and cot x
E. The students will solve applications involving inverse functions
6.3 Solving Basic Trigonometric Equations
A. The students will use a graph to gain information about principal roots, roots in (0, 2π) or
(0°, 360°) and roots in Ʀ
B. The students will use inverse functions to solve trig equations for the principal root
C. The students will solve trig equations for roots in (0, 2π) or (0°, 360°)
D. The students will solve trig equations for roots in Ʀ
E. The students will solve trig equations using fundamental identities
F. The students will solve trig equations using graphing technology
6.4 Solving General Trigonometric Equations and Applications
A. The students will use additional algebraic techniques to solve trig equations
B. The students will solve trig equations using multiple angle, sum and difference, and sumto-product identities
C. The students will solve trig equations of the form Asin(Bx+ or –C) + or – D = k
D. The students will use a combination of skills to model and solve a variety of applications
Unit Seven: Applications of Trigonometry ( weeks)
7.1 Oblique Triangles and the Law of Sines
A. The students will develop the law of sines and use it to solve ASA and AAS triangles
B. The students will solve SSA triangles (the ambiguous case) using the law of sines
C. The students will use the law of sines to solve applicants
7.2 The Law of Cosines and the Area of a Triangle
A. The students will apply the law of cosines when two sides and an included angle are
known (SAS)
B. The students will apply the law of cosines when three sides are known (SSS)
C. The students will solve applications using the law of cosines
D. The students will use trigonometry to find the area of a triangle
7.3 Vectors and Vector Diagrams
A. The students will represent a vector quantity geometrically
B. The students will represent a vector quantity graphically
C. The students will perform defined operations on vectors
D. The students will represent a vector quantity algebraically and find unit vectors
E. The students will use vector diagrams to solve applicants
7.4 Vector Applications and the Dot Product
A. The students will use vectors to investigate forces in equilibrium
B. The students will find the components of one vector along another
C. The students will solve applicants involving work
D. The students will compute dot products and the angle between two vectors
E. The students will find the projection of one vector along another and resolve a vector into
orthogonal components
F. The students will use vectors to develop an equation for nonvertical projectile motion and
solve related applications
Unit Eight: Trigonometric Connections to Algebra ( weeks)
8.1 Complex Numbers
A. The students will identify and simplify imaginary and complex numbers
B. The students will add and subtract complex numbers
C. The students will multiply complex numbers and find powers of i
D. The students will divide complex numbers
8.2 Complex Numbers in Trigonometric Form
A. The students will graph a complex number
B. The students will write a complex number in trigonometric form
C. The students will convert from trigonometric form to rectangular form
D. The students will interpret products and quotients geometrically
E. The students will compute products and quotients in trigonometric form
F. The students will solve applications involving complex numbers (optional)
8.3 De Moivre’s Theorem and the Theorem on nth Roots
A. The students will use De Moivre’s theorem to raise complex numbers to any power
B. The students will use De Moivre’s theorem to check solutions to polynomial equations
C. The students will use the nth roots theorem to find the nth roots of a complex number
8.4 Polar Coordinates, Equations, and Graphs
A. The students will plot points given in polar form
B. The students will express a point in polar form
C. The students will convert between polar and rectangular form
D. The students will sketch basic polar graphs using an r-value analysis
E. The students will use symmetry and families of curves to write a polar equation given a
polar graph or information about the graph
8.5 Parametric Equations and Their Graphs
A. The students will sketch the graph of a parametric equation
B. The students will write parametric equations in rectangular form
C. The students will graph curves from the cycloid family
D. The students will solve applications involving parametric equations
Key Assignments
Each lecture on a new section is video-taped and posted on the school website for the students to
view at home. In addition, a brief homework assignment is given with each lesson for students to
complete at home, usually consisting of about five to ten problems. Each assignment is designed
to reinforce the objectives learned in the video, and to check for understanding. All answers
must be justified with work. No credit will be awarded for the problems that are missing work.
Homework is reviewed the next day, and students are encouraged to raise questions as well as
answer each other’s questions. If all of the students’ questions are answered to their satisfaction
on any given day, they will work together on problems of greater depth.
Students will perform in-class activities throughout the year to reinforce the concepts from the
previous night’s homework. The major project of the course, which will take place in the fourth
quarter, will consist of a graphic art project. Each student will graph any of the functions we
have studied throughout their entire math education on a 17” x 22” sheet of graph paper. The
evaluation is based equally on mathematical accuracy and artistic creativity. This gives the
students an opportunity to express their creativity in mathematics, as well as causing them to
think in a different direction than they have done throughout the course. Rather than going from
a given equation to a graph, the student must compose a basic idea, and then find the equations
that best fit their idea.
Instructional Methods and or Strategies
Each day after school, the students watch a video at home of 20 to 30 minutes in length. They
then work on 5 to 10 homework problems, which should take an additional 20 to 30 minutes.
The next class period, we review each problem.
At the end of each chapter, I assign homework to review all the major concepts in the chapter.
There are generally two class periods between the last new concepts and the test. The first class
period is dedicated to answering the students’ questions on anything in the chapter, and the
second class period is dedicated to playing a review game with the Classroom Performance
System. I put various problems on the screen, and the students key in their answers on a remote
keypad. Their answers are then displayed on the screen, and a point is awarded to the team with
the most correct answers. At the end of the period, the team with the most points wins extracredit points.
Assessments including Methods and/or Tools
Students’ grades are calculated each semester using a weighted system. Students will be
informally assessed every day and monitored for understanding throughout all in-class activities.
Students will be formally assessed in their daily homework (12%), quizzes (12%), the major
project (equivalent to one chapter test), the chapter tests (56%), and the final exam (20%).