UNIT 3 – TRIGONOMETRIC FUNCTIONS UNIT 3 – TRIGONOMETRIC FUNCTIONS .................................................................................................................................. 1 ESSENTIAL CONCEPTS OF TRIGONOMETRY ......................................................................................................................... 4 INTRODUCTION................................................................................................................................................................................. 4 WHAT IS TRIGONOMETRY? ............................................................................................................................................................... 4 WHY TRIANGLES? ............................................................................................................................................................................ 4 EXAMPLES OF PROBLEMS THAT CAN BE SOLVED USING TRIGONOMETRY ........................................................................................... 4 GENERAL APPLICATIONS OF TRIGONOMETRIC FUNCTIONS ................................................................................................................ 4 WHY TRIGONOMETRY WORKS.......................................................................................................................................................... 4 EXTREMELY IMPORTANT NOTE ON THE NOTATION OF TRIGONOMETRY ............................................................................................. 5 TRIGONOMETRY OF RIGHT TRIANGLES – TRIGONOMETRIC RATIOS OF ACUTE ANGLES ...................................................................... 5 THE SPECIAL TRIANGLES – TRIGONOMETRIC RATIOS OF SPECIAL ANGLES ........................................................................................ 5 HOMEWORK ..................................................................................................................................................................................... 6 IN-CLASS PRACTICE ......................................................................................................................................................................... 6 SUMMARY: ESSENTIAL IDEAS OF TRIGONOMETRY ........................................................................................................... 7 TRIGONOMETRY RELATES ANGLES TO DISTANCES ............................................................................................................................ 7 TRIGONOMETRIC RELATIONSHIPS IN TRIANGLES ............................................................................................................................... 7 EXAMPLES OF TRIGONOMETRIC RELATIONSHIPS IN THE PHYSICAL WORLD ....................................................................................... 7 RADIAN MEASURE ......................................................................................................................................................................... 8 SUMMARY OF VARIOUS UNITS FOR MEASURING ANGLES .................................................................................................................. 8 CALCULATOR USE ............................................................................................................................................................................ 8 WHY 360 DEGREES IN ONE FULL REVOLUTION? ............................................................................................................................... 8 DEFINITION OF THE RADIAN ............................................................................................................................................................. 9 INVESTIGATION – THE RELATIONSHIP AMONG , R AND L .................................................................................................................. 9 A MORE ANALYTICAL APPROACH TO FINDING THE RELATIONSHIP AMONG , R AND L ..................................................................... 10 How many Radians are there in One Full Revolution? .............................................................................................................. 10 How is related to the “Amount of Rotation?” ......................................................................................................................... 10 How is l related to the Circumference of a Circle? .................................................................................................................... 10 SUMMARY: CALCULATING THE LENGTH OF AN ARC ........................................................................................................................ 11 Cr 2 IS A SPECIAL CASE OF lr ........................................................................................................................................ 11 CONVERTING BETWEEN RADIANS AND DEGREES............................................................................................................................. 11 Examples................................................................................................................................................................................... 11 Special Angles ........................................................................................................................................................................... 11 RADIANS ARE DIMENSIONLESS ....................................................................................................................................................... 12 ANGULAR FREQUENCY (ANGULAR SPEED) ..................................................................................................................................... 12 Example 1 ................................................................................................................................................................................. 12 Solution ..................................................................................................................................................................................... 12 Example 2 ................................................................................................................................................................................. 12 Solution ..................................................................................................................................................................................... 12 EXAMPLE 3 .................................................................................................................................................................................... 13 Solution ..................................................................................................................................................................................... 13 HOMEWORK ................................................................................................................................................................................... 13 RADIAN MEASURE AND ANGLES ON THE CARTESIAN PLANE ........................................................................................ 14 TRIGONOMETRY OF RIGHT TRIANGLES – TRIGONOMETRIC RATIOS OF ACUTE ANGLES .................................................................... 14 THE SPECIAL TRIANGLES – TRIGONOMETRIC RATIOS OF SPECIAL ANGLES ...................................................................................... 14 TRIGONOMETRIC RATIOS OF ANGLES OF ROTATION – TRIGONOMETRIC RATIOS OF ANGLES OF ANY SIZE ........................................ 14 COTERMINAL ANGLES .................................................................................................................................................................... 15 PRINCIPAL ANGLE .......................................................................................................................................................................... 16 EXAMPLE – EVALUATING TRIG RATIOS BY USING THE RELATED FIRST QUADRANT ANGLE (REFERENCE ANGLE) ............................ 16 QUESTION ...................................................................................................................................................................................... 17 Answer ...................................................................................................................................................................................... 17 ADDITIONAL TOOLS FOR DETERMINING TRIG RATIOS OF SPECIAL ANGLES ..................................................................................... 17 The Unit Circle ......................................................................................................................................................................... 17 The Rule of Quarters (Beware of Blind Memorization!)............................................................................................................. 17 HOMEWORK ................................................................................................................................................................................... 18 Copyright ©, Nick E. Nolfi MCR3U9 Unit 3 – Trigonometric Functions TF-1 INTRODUCTION TO TRIGONOMETRIC FUNCTIONS .......................................................................................................... 19 OVERVIEW ....................................................................................................................................................................................