Unit Title: Introduction To The Unit Circle, Radians, & Trigonometric Functions Name: Steven K. Content Area: Algebra II / Trigonometry Grade Level: 9-12 CA Content Standard(s)/Common Core Standard(s): TRIGONOMETRIC FUNCTIONS (F-TF): Extend the domain of trigonometric functions using the unit circle. 1. Understand the radian measure of an angle as the length of the arc on the unit circle subtended by the angle. 2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. 2.1. Graph all 6 basic trigonometric functions. Model periodic phenomena with trigonometric functions. 5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Prove and apply trigonometric identities. Big Ideas: RADIANS - A unit of measure that we can use instead of degrees in order to measure angles. THE UNIT CIRCLE - A circle of radius = 1, centered at the origin - the basis on which we build and study trigonometry. TRIGONOMETRIC FUNCTIONS - We will study 6 main trig. functions: sin(x), cos(x), tan(x), csc(x), sec(x), & cot(x). INVERSE TRIG. FUNCTIONS - We will study inverse functions for each of the 6 trig. functions we study. GRAPHING TRIG. FUNCTIONS - We will learn and study graphs of trig. functions including transformations. VECTORS - We will apply our knowledge of trig. to the study of vectors REAL-WORLD PHENOMENA - We will use trig to study real-world scenarios.
Unit Goals and Objectives: Students will learn to solve for fractions of circumference ("slices of pie") as proportions of the full circumference with a given degree or radian measure. Students will learn the true meaning of "pi", what radians are, and how to freely convert between degree and radian measures. Students will learn, graph, and apply the 6 basic trig. functions: sin(x), cos(x), tan(x), csc(x), sec(x), cot(x). Students will learn, graph, and apply the 6 inverse trig. functions Students will be able to identify graphs of trig functions and their inverses. Students will be able to apply transformations to trigonometric equations and graphs. Students will learn what the unit circle is, why it is important to the study of trig., and will begin memorizing "important angles" from the unit circle (all multiples of 0/30/45/60/90/360 degrees) including right triangle ratios, and values of trig. functions at each of these radian/degree values. Students will learn what vectors are, how to break them into x/y components, and how to add and subtract vectors using trig. functions. Students will apply trig. to real world phenomena. They will be able to identify natural and unnatural phenomena that can be modeled with the trig. they have learned. Students will be able to generate and solve for their own made-up real-world scenarios that utilize trigonometry. Unit Summary: In this unit we will introduce concepts including: the unit circle, radians, trigonometric functions, as well as their associated graphs. Building upon the foundations of Algebra and Geometry, students will learn to connect this understanding to new trigonometric concepts - thus expanding overall mathematical knowledge. This will most likely be students' first formal exposure with trigonometry. Therefore, concepts will be introduced from the ground up in order to build students' foundational trigonometry skills which will allow them to excel broadly in future math courses.
Assessment Plan: Entry-Level: 1. Guided Notes: Examples to find circumference (C) and fractions of circumference (C). 2. Guided Notes: What is pi? Why do we use radians? 3. Guided Notes: Example problems to convert degrees radians. Formative: 1. Randomized Student Questions 2. Fill In The Blank Unit Circle 3. Trig. Functions Worksheet I 4. Quiz: Unit Circle, Pi, Radians, & Basic Trig. Func. 5. Trig. Functions Worksheet II 6. Graphing Trig. Functions Worksheet 7. Graphing Trig. Functions Group Worksheet 8. Quizlet: Flashcards & Take Home Quiz Summative: 1. Trigonometry Graphs Art Project 2. Trigonometry Art Project Presentation 3. Unit Exam Lesson 1 1. Students will learn to solve for a fraction of circumference with a given angle measure. 2. Students will learn what pi really is. 3. Students will learn what a radian is. 4. Students will learn what the unit circle is. 5. Students will learn to convert between radian degree. 1. Students will be able to solve for circumference and fraction of circumference in example problems. 2. Students will know that pi is the number of radians in half a circle (180 degrees). 3. Students will be able to solve for radian measures given a degree measure by taking the fraction of 180 degrees/pi. Students will see the connection between multiplying by a fraction to get a part of circumference, and multiplying by a fraction. X Communication X Collection Organization X Interaction 1. Teacher Lecture 2. Example Problems i.) Independent in front of class ii.) Group work 4. Students will begin to be able to fill in unit circle values. 5. Students will be able to solve example problems and freely convert between degree radians.
Lesson 2 1. Students will know how to utilize the Pythagorean theorem. 2. Students will know the 6 basic trig. functions and the ratios that they represent. 3. Students will be able to solve for unknown sides of triangles given the value of one or more of its trig. functions. 1. Students will solve for unknown sides of right triangles using the Pythagorean theorem. 2. Students will be able to identify which sides each trig ratio utilizes, and how. Students will be able to solve for the value of trig. functions given a triangle with some known sides. Communication Collection X Organization X Interaction 1. Webercise w/ included videos 3. Students will be able to solve for unknown sides of triangles and the values of other trig. functions, given the value of one or more trig. functions in a given triangle. Lesson 3 1. Students will be able to fill in all important values on the unit circle. This includes important angle measures, trig. ratios, coordinates along the edge of the unit circle, and other values. 1. Students will completely fill in the unit circle graphic organizer worksheet with all appropriate values. X Communication X Collection X Organization Interaction 1. Unit Circle Worksheet/Graphic Organizer
Unit Resources: Trigonometry Identity Reference & Cheat Sheet - by: Lamar University (http://tutorial.math.lamar.edu/pdf/trig_cheat_sheet.pdf) Trigonometry Identity Reference & Cheat Sheet - by: UC Irvine (http://www.math.uci.edu/~etiti/teaching/math-2b/trig_cheat_sheet_reduced.pdf) Useful Websites: Basic Trigonometry - by: Khan Academy https://www.youtube.com/watch?feature=player_embedded&v=jsiy4txgime Basic Trigonometry II - by: Khan Academy https://www.youtube.com/watch?feature=player_embedded&v=g-t_6hcdmqc Radians & Degrees, The Unit Circle, & Trig. Functions - by: Khan Academy https://www.youtube.com/watch?feature=player_embedded&v=9zspw8u6kqm Using Trig. Functions - by: Khan Academy https://www.youtube.com/watch?feature=player_embedded&v=znr9tw4aizi Using Trig. Functions Part II - by: Khan Academy https://www.youtube.com/watch?feature=player_embedded&v=roxmkyjplgk The Unit Circle Definition of Trig. Functions - by: Khan Academy https://www.youtube.com/watch?feature=player_embedded&v=6qv_bplqs8e The Unit Circle Definition of Trig. Functions Part II - by: Khan Academy https://www.youtube.com/watch?feature=player_embedded&v=zffzvsh285c Graph of the Sine Function - by: Khan Academy https://www.youtube.com/watch?feature=player_embedded&v=2zoiw4pdvko Graphs of Trig. Functions - by: Khan Academy https://www.youtube.com/watch?feature=player_embedded&v=qmxmppkzpme Graphing Trig. Functions - by: Khan Academy https://www.youtube.com/watch?feature=player_embedded&v=vhyi93uv5kg More Trig. Graphs - by: Khan Academy https://www.youtube.com/watch?feature=player_embedded&v=nig3l8owkye Determining The Equation of a Trigonometric Function - by: Khan Academy https://www.youtube.com/watch?feature=player_embedded&v=yvw5l9w1hge Trigonometric Identities - by: Khan Academy https://www.youtube.com/watch?feature=player_embedded&v=olzxqiqzzz0