TRIGONOMETRY GRADES 11-12

THE EWING PUBLIC SCHOOLS 2099 Pennington Road Ewing, NJ 08618

Board Approval Date: October 29, 2012 Michael Nitti Written by: EHS Department Superintendent

In accordance with The Ewing Public Schools’ Policy 2230, Course Guides, this curriculum has been reviewed and found to be in compliance with all policies and all affirmative action criteria.

Table of Contents

Page

Scope of Essential Learning:

Unit 1: (10 Days) 1

Unit 2: Analytic Trigonometry (15 Days) 3

Unit 3: Laws and Vectors (13 Days) 5

Unit 4: Complex (13 Days) 7

Unit 5: Exponential and Logarithmic Numbers (13 Days) 10

Unit 6: Topics in Analytic Geometry (12 Days) 12

1

Unit 1: Trigonometry (10 Days)

Why Is This Unit Important?

In this unit, students will learn ways to evaluate and use and their inverses to solve real-life problems involving right triangles, directional bearings and harmonic motion.

Enduring Understandings: Students will understand:

• How to describe an angle and convert between radian and degree measure • How to identify a and its relationship to real numbers • How to evaluate trigonometric functions of any angle • How to use the fundamental trigonometric identities • How to sketch the graph of trigonometric functions and translations of graphs of and cosine functions • How to evaluate the inverse trigonometric functions • How to evaluate the composition of trigonometric functions and inverse trigonometric functions

Essential Questions:

• Differentiate between radians and degrees: How do they differ? How are they similar? • Why is it called a unit circle? • What do trigonometric functions reveal?

Acquired Knowledge: After studying the material of this unit, students will be able to:

• Describe angles • Identify a unit circle and its relationship to real numbers • Evaluate trigonometric functions using the unit circle • Evaluate trigonometric functions of acute angles • Evaluate trigonometric functions of any angles • Evaluate trigonometric functions of real numbers • Evaluate the inverse sine function • Evaluate the other inverse trigonometric functions • Evaluate the compositions of trigonometric functions

Acquired Skills: After studying the material of this unit, the student should be able to:

• Use radian measure • Use degree measure • Use the domain and period to evaluate sine and cosine functions 2

• Use fundamental trigonometric identities • Use reference angles to evaluate trigonometric functions • Sketch the graphs of basic sine and cosine functions • Use amplitude and period to sketch the graphs of sine and cosine functions • Sketch translations of graphs of sine and cosine functions • Sketch the graphs of tangent functions • Sketch the graphs of cotangent functions • Sketch the graphs of secant and cosecant functions • Sketch the graphs of damped trigonometric functions

Differentiation:

Enrichment : • Presentation on Applications of Inverse Functions

Supplement : • Radian and Degree Game • Unit Circle Activity

Major Assessments:

• Quizzes • Unit Test

List of Applicable CCSS and Standards/CPIs Covered in This Unit:

F-TF.1-9 G-C.1-5 MP.1-8

Suggested Learning Experiences and Instructional Activities:

• Estimate the radian • Identify That Identity! • Sketch translations • Damp That Function! 3

Unit 2: Analytic Trigonometry (15 Days)

Why Is This Unit Important?

In this unit, students will learn how to use fundamental trigonometric identities to evaluate functions, simplify expressions, develop additional identities and solve equations.

Enduring Understandings: Students will understand:

• How to use fundamental trigonometric identities to evaluate trigonometric functions and simplify trigonometric expressions • How to verify trigonometric identities • How to use standard algebraic techniques and inverse trigonometric functions to solve trigonometric equations • How to use sum and difference formulas, and multiple angle formulas, power- reducing formulas, half-angle formulas, and product-to-sum and sum-to-product formulas to rewrite and evaluate trigonometric functions

Essential Questions:

• Why are relations and functions represented in multiple ways? • How are the properties of functions and functional operations useful?

Acquired Knowledge: After studying the material of this unit, students will be able to:

• Recognize and write the fundamental trigonometric identities • Verify trigonometric identities • Know the following formulas: ◊ sum and difference ◊ multiple angle ◊ power-reducing ◊ half-angle ◊ product-to-sum ◊ sum-to-product

Acquired Skills: After studying the material of this unit, the student should be able to:

• Use the fundamental trigonometric identities to evaluate trigonometric functions, and simplify and rewrite trigonometric expressions • Plan a strategy for verifying trigonometric identities • Solve trigonometric equations of quadratic type • Solve trigonometric equations involving multiple angles

4

Differentiation:

Enrichment : • Multiple Angle Challenges

Supplement : • Evaluate  Simplify  Rewrite

Major Assessments:

• Quizzes • Unit Test

List of Applicable CCSS and Standards/CPIs Covered in This Unit:

F-TF.1-9 MP.1-8

Suggested Learning Experiences and Instructional Activities:

• Identity or Formula? • Identify That Identity! • Which Formula Best Applies?

5

Unit 3: Laws and Vectors (13 Days)

Why Is This Unit Important?

This unit will serve to develop the skills for representing and solving a wide range of situations and problems that involve any quantities that have a directional component associated with them.

Enduring Understandings: Students will understand:

• Trigonometric laws can be utilized to solve for unknown sides, angles and areas of oblique triangles • How to use vector operations and dot products to represent and solve real-world problems involving directional quantities such as force and momentum

Essential Questions:

• Why are functions and relations represented by vectors?

Acquired Knowledge: After studying the material of this unit, students will be able to:

• Know the following formulas: ◊ Law of ◊ Law of Cosines ◊ Heron’s Area Formula • Represent vectors as directed line segments • Use the Properties of the Dot Product

Acquired Skills: After studying the material of this unit, the student should be able to:

• Use the Law of Sines to solve oblique triangles • Use the Law of Cosines to solve oblique triangles • Find the area of an oblique triangle • Use Heron’s Area formula to find the area of a triangle • Represent vector operations graphically • Write component form of vectors • Perform basic vector operations • Find the direction angles of vectors • Find the angle between two vectors • Write vectors as linear combinations of unit vectors • Find the dot product of two vectors • Determine of two vectors are orthogonal • Write a vector as the sum of two vector components

6

Differentiation:

Enrichment : • Physics Applications

Supplement : • Graphical Vector Solutions – Doing it by Scale

Major Assessments:

• Quizzes • Unit Test

List of Applicable CCSS and Standards/CPIs Covered in This Unit:

F-TF.8-9 MP.1-8 N-VM.1-5

Suggested Learning Experiences and Instructional Activities:

• Law of Sines or Cosine? • Applications of Heron’s Area Formula • Graphing Vector Scenarios • Determining Direction? • Dot Products

7

Unit 4: Complex Numbers (13 Days)

Why Is This Unit Important?

Sometimes, in mathematical processes, equations arise with no real solution. To move past such situations, mathematicians expanded the system to include imaginary numbers based upon the basic unit of the root of -1 ( i). When real numbers are added to imaginary numbers, complex numbers are formed. In this unit, students will learn how to apply mathematical techniques to complex numbers to move past equations with no real solutions back into mathematics where real solutions form.

Enduring Understandings: Students will understand:

• How to perform operations with complex numbers • How to determine the number of zeroes of functions • How to multiply and divide complex numbers written in trigonometric form • How to find powers and nth roots of complex numbers

Essential Questions:

• What makes a number ‘complex’? • Why use ‘imaginary’ numbers?

Acquired Knowledge: After studying the material of this unit, students will be able to:

• Determine the number of solutions of polynomial equations • Determine the trigonometric form of complex numbers • Know the following theorems: ◊ The Fundamental Theorem of Algebra ◊ Linear Theorem ◊ DeMoivre’s Theorem

Acquired Skills: After studying the material of this unit, the student should be able to:

• Use the imaginary unit i to write complex numbers • Add, subtract and multiply complex numbers • Use the to write the quotient of two complex numbers in standard form • Find complex solutions of quadratic equations • Find solutions of polynomial equations • Multiply and divide complex numbers written in trigonometric form • Use DeMoivre’s Theorem to find powers of complex numbers

8

Differentiation:

Enrichment : • Presentation on Applications of Complex Numbers

Supplement : • Uses of i

Major Assessments:

• Quizzes • Unit Test

List of Applicable CCSS and Standards/CPIs Covered in This Unit:

FMP.1-8 N-CN.1-9

Suggested Learning Experiences and Instructional Activities:

• Using the complex conjugates • Complex solutions • Plot complex numbers in the

9

Unit 5: Exponential and Logarithmic Functions (13 Days)

Why Is This Unit Important?

Mathematics is a powerful tool for representing, describing, interpreting and evaluating real-world phenomena. Many such phenomena are not typically static or follow a constant linear growth or decline. Many of these situations found in the real world follow an exponential or logarithmic growth, decay or a cycle involving both. In this unit, students will learn strategies and techniques that will permit them to represent, describe, interpret and evaluate such situations.

Enduring Understandings: Students will understand:

• Use exponential growth models, exponential decay models and logarithmic models to solve real-life problems

Essential Questions:

• How do exponential functions model real-world problems and their solutions? • How do logarithmic functions model real-world problems and their solutions?

Acquired Knowledge: After studying the material of this unit, students will be able to:

• Recognize and evaluate exponential and logarithmic functions • Recognize the five most common types of models involving exponential and logarithmic functions: ◊ Exponential growth model ◊ Exponential decay model ◊ Gaussian model ◊ Logistic growth model ◊ Logarithmic models

Acquired Skills: After studying the material of this unit, the student should be able to:

• Use the change-of-base formula to rewrite and evaluate logarithmic expressions • Use properties of to evaluate, rewrite, expand or condense logarithmic expressions • Solve exponential and logarithmic equations

Differentiation:

Enrichment : • Reporting on Real-Life Gaussian Fluctuations

10

Supplement : • Reporting on Real-Life Growth and Decay Situations

Major Assessments:

• Quizzes • Unit Test

List of Applicable CCSS and Standards/CPIs Covered in This Unit:

F-LE.1-5 MP.1-8

Suggested Learning Experiences and Instructional Activities:

• Graph exponential and logarithmic functions • Develop scenarios for exponential and logarithmic models

11

Unit 6: Topics in Analytic Geometry (12 Days)

Why Is This Unit Important?

In mathematics, it is fairly simple to apply algebraic and geometric operations to represent and solve for situations. However, such processes are generally limited to two variables, such as vertical and horizontal distance from an origin point. While such situations often arise, and algebra and geometry provide the mathematics to deal with such cases, in the real world there are often third variables to be considered.

For example, for a projectile, the time it is at a set vertical height and horizontal distance may be critical; i.e., a player catching a ball must be not just at the point a ball is in the air, but also at the time it returns to the ground. The same can be said for intercepting an incoming missile.

Another aspect of real-life phenomena transcending normal mathematical procedure is the origin point of revolving objects. While it is easier to graphically plot the origin at the center, there are times where objects are better analyzed at one of the foci points; i.e., our Sun-Earth system.

In this unit, students will apply trigonometric techniques to analytical geometry to solve for such real-life situations.

Enduring Understandings: Students will understand:

• How to find the inclination of a line, the angle between two lines, and the distance between a point and a line • How to write the standard form of the equation of a parabola, ellipse and a hyperbola • How to eliminate the xy-term in the equation of a conic and use the discriminant to identify a conic • How to rewrite a set of parametric equations as a rectangular equation and find a set of parametric equations for a graph • How to write equations in polar form and graph polar equations

Essential Questions:

• Why are functions and relations represented by parametric equations? • Why are functions and relations represented by polar equations?

Acquired Knowledge: After studying the material of this unit, students will be able to:

• Recognize a conic as the intersection of a plane and a double-napped cone • Understand the reflective property of parabolas • Understand properties of ellipses 12

• Understand properties of hyperbolas • Evaluate a set of parametric equations for a given value of the parameter • Recognize special polar graphs • Define conics in terms of eccentricity

Acquired Skills: After studying the material of this unit, the student should be able to:

• Find the eccentricity of an ellipse • Find the asymptotes of a hyperbola • Classify a conic from its general equations • Plot points on a the • Write equations of conics in polar form

Differentiation:

Enrichment : • Parabolic Equations in Time

Supplement : • Kepler’s Law

Major Assessments:

• Quizzes • Unit Test

List of Applicable CCSS and Standards/CPIs Covered in This Unit:

G-CO.9-11 G-GMO.1-4 G-GPE.1-7 MP.1-8

Suggested Learning Experiences and Instructional Activities:

• Find the equation of a tangent line to a parabola at a given point • Sketch the curve that is represented by a set of parametric equations • Convert points from rectangular to polar form and vice versa • Use symmetry, zeros and maximum r-values to sketch graphs of polar equations