TABLE OF CONTENTS - UNIT 2 – TRIGONOMETRIC FUNCTIONS TABLE OF CONTENTS - UNIT 2 – TRIGONOMETRIC FUNCTIONS ........................................................................................ 1 ESSENTIAL CONCEPTS OF TRIGONOMETRY ............................................................................................................................. 3 WHAT IS TRIGONOMETRY? .................................................................................................................................................................... 3 WHY TRIANGLES? ................................................................................................................................................................................. 3 EXAMPLES OF PROBLEMS THAT CAN BE SOLVED USING TRIGONOMETRY .............................................................................................. 3 GENERAL APPLICATIONS OF TRIGONOMETRIC FUNCTIONS .................................................................................................................... 3 EXTREMELY IMPORTANT PREREQUISITE KNOWLEDGE .......................................................................................................................... 3 TRIGONOMETRY OF RIGHT TRIANGLES – TRIGONOMETRIC RATIOS OF ACUTE ANGLES ........................................................................ 4 THE SPECIAL TRIANGLES – TRIGONOMETRIC RATIOS OF SPECIAL ANGLES ........................................................................................... 4 HOMEWORK ........................................................................................................................................................................................... 4 TRIGONOMETRIC RATIOS OF ANGLES OF REVOLUTION (ROTATION) – TRIGONOMETRIC RATIOS OF ANGLES OF ANY SIZE .................. 4 COTERMINAL ANGLES ........................................................................................................................................................................... 5 EXAMPLE ............................................................................................................................................................................................... 6 Question ........................................................................................................................................................................................... 6 Answer .............................................................................................................................................................................................. 6 ADDITIONAL TOOLS FOR DETERMINING TRIG RATIOS OF SPECIAL ANGLES .......................................................................................... 6 The Unit Circle ................................................................................................................................................................................. 6 The Rule of Quarters (Beware of Blind Memorization!) .................................................................................................................. 6 EXERCISE ............................................................................................................................................................................................... 7 HOMEWORK ........................................................................................................................................................................................... 7 LAW OF SINES AND LAW OF COSINES .......................................................................................................................................... 8 LAW OF COSINES ................................................................................................................................................................................... 8 LAW OF SINES ........................................................................................................................................................................................ 8 IMPORTANT EXERCISE ........................................................................................................................................................................... 8 EXAMPLES ............................................................................................................................................................................................. 8 HOMEWORK ......................................................................................................................................................................................... 10 PROOFS OF THE PYTHAGOREAN THEOREM, THE LAW OF COSINES AND THE LAW OF SINES ............................. 11 INTRODUCTION .................................................................................................................................................................................... 11 NOLFI’S INTUITIVE DEFINITION OF “PROOF” ....................................................................................................................................... 11 PYTHAGOREAN THEOREM EXAMPLE ................................................................................................................................................... 11 Proof: ............................................................................................................................................................................................. 11 PROOF OF THE LAW OF COSINES .......................................................................................................................................................... 12 PROOF OF THE LAW OF SINES ............................................................................................................................................................... 12 IMPORTANT QUESTIONS – TEST YOUR UNDERSTANDING ..................................................................................................................... 13 THE RECIPROCAL TRIGONOMETRIC RATIOS ........................................................................................................................ 14 REVIEW ................................................................................................................................................................................................ 14 WHY ON EARTH DO WE NEED MORE RATIOS? ...................................................................................................................................... 14 EXAMPLES ........................................................................................................................................................................................... 14 EXERCISES ........................................................................................................................................................................................... 15 APPLICATIONS OF SIMPLE GEOMETRY AND TRIGONOMETRIC RATIOS ...................................................................... 16 TRIGONOMETRIC IDENTITIES ...................................................................................................................................................... 20 REVIEW – TYPES OF EQUATIONS STUDIED THUS FAR .......................................................................................................................... 20 WHAT IS AN IDENTITY? ........................................................................................................................................................................ 20 LIST OF IDENTITIES THAT WE ALREADY KNOW ................................................................................................................................... 20 IMPORTANT NOTE ABOUT NOTATION .................................................................................................................................................. 20 EXAMPLES ........................................................................................................................................................................................... 21 LOGICAL AND NOTATIONAL PITFALLS – PLEASE AVOID ABSURDITIES! .............................................................................................. 21 SUGGESTIONS FOR PROVING TRIG IDENTITIES ..................................................................................................................................... 21 HOMEWORK ......................................................................................................................................................................................... 21 INTRODUCTION TO TRIGONOMETRIC FUNCTIONS .............................................................................................................. 22 OVERVIEW ........................................................................................................................................................................................... 22 GRAPHS ............................................................................................................................................................................................... 22 Questions about the Sine Function ................................................................................................................................................. 22 Copyright ©, Nick