0501 ch 5 day 1

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  • 0501 ch 5 day 1

    1. 1. Chapter 5Trigonometric Functions of Real Numbers
    2. 2. Chapter 5Trigonometric Functions of Real Numbers 5.1 The Unit Circle
    3. 3. Chapter 5Trigonometric Functions of Real Numbers 5.1 The Unit Circle John 3:16 For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life.
    4. 4. Assumptions
    5. 5. Assumptions1. Much of this chapter is review for you
    6. 6. Assumptions1. Much of this chapter is review for you2. You will ask questions about concepts you don’t understand or skills you can’t do or remember
    7. 7. Assumptions1. Much of this chapter is review for you2. You will ask questions about concepts you don’t understand or skills you can’t do or remember3. You have your Unit Circle with you in in class and when doing homework
    8. 8. Two Approaches to Trigonometry
    9. 9. Two Approaches to Trigonometry1. Unit Circle (Chapter 5)
    10. 10. Two Approaches to Trigonometry1. Unit Circle (Chapter 5)2. Right Triangle (Chapter 6)
    11. 11. Two Approaches to Trigonometry1. Unit Circle (Chapter 5)2. Right Triangle (Chapter 6)we work with both as each has its strengths
    12. 12. Two Approaches to Trigonometry1. Unit Circle (Chapter 5)2. Right Triangle (Chapter 6)we work with both as each has its strengths Note: In this chapter we will use mostly radians; in Chapter 6 we will use mostly degrees
    13. 13. The Unit Circle is the circle of radius 1 centered at the origin
    14. 14. The Unit Circle is the circle of radius 1 centered at the origin 2 2 x + y =1
    15. 15. ⎛ 6 7 ⎞Is the point ⎜ 6 , 6 ⎟ on the Unit Circle? ⎝ ⎠
    16. 16. ⎛ 6 7 ⎞Is the point ⎜ 6 , 6 ⎟ on the Unit Circle? ⎝ ⎠ 2 2 x + y =1
    17. 17. ⎛ 6 7 ⎞Is the point ⎜ 6 , 6 ⎟ on the Unit Circle? ⎝ ⎠ 2 2 x + y =1 2 2 ⎛ 6 ⎞ ⎛ 7 ⎞ ⎜ 6 ⎟ + ⎜ 6 ⎟ = 1 ⎝ ⎠ ⎝ ⎠
    18. 18. ⎛ 6 7 ⎞Is the point ⎜ 6 , 6 ⎟ on the Unit Circle? ⎝ ⎠ 2 2 x + y =1 2 2 ⎛ 6 ⎞ ⎛ 7 ⎞ ⎜ 6 ⎟ + ⎜ 6 ⎟ = 1 ⎝ ⎠ ⎝ ⎠ 6 7 + =1 36 36
    19. 19. ⎛ 6 7 ⎞Is the point ⎜ 6 , 6 ⎟ on the Unit Circle? ⎝ ⎠ 2 2 x + y =1 2 2 ⎛ 6 ⎞ ⎛ 7 ⎞ ⎜ 6 ⎟ + ⎜ 6 ⎟ = 1 ⎝ ⎠ ⎝ ⎠ 6 7 + =1 36 36 13 ≠1 36 No
    20. 20. ⎛ 35 ⎞Point P ⎜ , y⎟ is on the Unit Circle in quadrant IV. ⎝ 6 ⎠ Find y .
    21. 21. ⎛ 35 ⎞Point P ⎜ , y⎟ is on the Unit Circle in quadrant IV. ⎝ 6 ⎠ Find y . 2⎛ 35 ⎞ 2⎜ 6 ⎟ + y = 1⎝ ⎠
    22. 22. ⎛ 35 ⎞Point P ⎜ , y⎟ is on the Unit Circle in quadrant IV. ⎝ 6 ⎠ Find y . 2⎛ 35 ⎞ 2⎜ 6 ⎟ + y = 1⎝ ⎠ 35 2 36 +y = 36 36
    23. 23. ⎛ 35 ⎞Point P ⎜ , y⎟ is on the Unit Circle in quadrant IV. ⎝ 6 ⎠ Find y . 2⎛ 35 ⎞ 2⎜ 6 ⎟ + y = 1⎝ ⎠ 35 2 36 +y = 36 36 2 1 y = 36
    24. 24. ⎛ 35 ⎞Point P ⎜ , y⎟ is on the Unit Circle in quadrant IV. ⎝ 6 ⎠ Find y . 2⎛ 35 ⎞ 2⎜ 6 ⎟ + y = 1⎝ ⎠ 35 2 36 +y = 36 36 2 1 y = 36 1 y=± 36
    25. 25. ⎛ 35 ⎞Point P ⎜ , y⎟ is on the Unit Circle in quadrant IV. ⎝ 6 ⎠ Find y . 2⎛ 35 ⎞ 2⎜ 6 ⎟ + y = 1⎝ ⎠ 35 2 36 +y = 36 36 2 1 y = 36 1 y=± 36 1 y=± 6
    26. 26. ⎛ 35 ⎞Point P ⎜ , y⎟ is on the Unit Circle in quadrant IV. ⎝ 6 ⎠ Find y . 2⎛ 35 ⎞ 2⎜ 6 ⎟ + y = 1⎝ ⎠ 1 We choose − as P is in Q IV 35 2 36 6 +y = 36 36 1 1 ∴ y=− 2 y = 6 36 1 y=± 36 1 y=± 6
    27. 27. When the initial ray rotates through some angle, θ , its ending position is calledthe terminal ray and the point of intersection of the terminal ray and the Unit Circle is the terminal point.
    28. 28. Key Terminal Points (Your Unit Circle)
    29. 29. Key Terminal Points (Your Unit Circle)
    30. 30. Key Terminal Points (Your Unit Circle)These values are generated by using the properties of30-60-90 and 45-45-90 triangles. You did this in AAT.
    31. 31. Find the terminal point if: 5π 1. θ = 4
    32. 32. Find the terminal point if: 5π 1. θ = 4 ⎛ 2 2 ⎞ ⎜ − 2 ,− 2 ⎟ ⎝ ⎠
    33. 33. Find the terminal point if: 5π 1. θ = 4 ⎛ 2 2 ⎞ ⎜ − 2 ,− 2 ⎟ ⎝ ⎠ π 2. θ = − 6
    34. 34. Find the terminal point if: 5π 1. θ = 4 ⎛ 2 2 ⎞ ⎜ − 2 ,− 2 ⎟ ⎝ ⎠ π 2. θ = − 6 12π π − 6 6
    35. 35. Find the terminal point if: 5π 1. θ = 4 ⎛ 2 2 ⎞ ⎜ − 2 ,− 2 ⎟ ⎝ ⎠ π 2. θ = − 6 12π π − 6 6 11π same as 6
    36. 36. Find the terminal point if: 5π 1. θ = 4 ⎛ 2 2 ⎞ ⎜ − 2 ,− 2 ⎟ ⎝ ⎠ π 2. θ = − 6 12π π − 6 6 11π same as 6 ⎛ 3 1 ⎞ ⎜ 2 ,− 2 ⎟ ⎝ ⎠
    37. 37. Find the terminal point if: 5π 1. θ = 4 ⎛ 2 2 ⎞ 7π ⎜ − 2 ,− 2 ⎟ 3. θ = − ⎝ ⎠ 6 π 2. θ = − 6 12π π − 6 6 11π same as 6 ⎛ 3 1 ⎞ ⎜ 2 ,− 2 ⎟ ⎝ ⎠
    38. 38. Find the terminal point if: 5π 1. θ = 4 ⎛ 2 2 ⎞ 7π ⎜ − 2 ,− 2 ⎟ 3. θ = − ⎝ ⎠ 6 π 12π 7π 2. θ = − − 6 6 6 12π π − 6 6 11π same as 6 ⎛ 3 1 ⎞ ⎜ 2 ,− 2 ⎟ ⎝ ⎠
    39. 39. Find the terminal point if: 5π 1. θ = 4 ⎛ 2 2 ⎞ 7π ⎜ − 2 ,− 2 ⎟ 3. θ = − ⎝ ⎠ 6 π 12π 7π 2. θ = − − 6 6 6 12π π − 5π 6 6 same as 6 11π same as 6 ⎛ 3 1 ⎞ ⎜ 2 ,− 2 ⎟ ⎝ ⎠
    40. 40. Find the terminal point if: 5π 1. θ = 4 ⎛ 2 2 ⎞ 7π ⎜ − 2 ,− 2 ⎟ 3. θ = − ⎝ ⎠ 6 π 12π 7π 2. θ = − − 6 6 6 12π π − 5π 6 6 same as 6 11π same as ⎛ 3 1 ⎞ 6 ⎜ − 2 , 2 ⎟ ⎝ ⎠ ⎛ 3 1 ⎞ ⎜ 2 ,− 2 ⎟ ⎝ ⎠
    41. 41. Find the terminal point if: 7π 4. θ = − 4
    42. 42. Find the terminal point if: 7π 4. θ = − 4 8π 7π − 4 4
    43. 43. Find the terminal point if: 7π 4. θ = − 4 8π 7π − 4 4 π same as 4
    44. 44. Find the terminal point if: 7π 4. θ = − 4 8π 7π − 4 4 π same as 4 ⎛ 2 2 ⎞ ⎜ 2 , 2 ⎟ ⎝ ⎠
    45. 45. Find the terminal point if: 7π 55π 4. θ = − 5. θ = 4 6 8π 7π − 4 4 π same as 4 ⎛ 2 2 ⎞ ⎜ 2 , 2 ⎟ ⎝ ⎠
    46. 46. Find the terminal point if: 7π 55π 4. θ = − 5. θ = 4 6 8π 7π 55π 48π − − 4 4 6 6 π same as 4 ⎛ 2 2 ⎞ ⎜ 2 , 2 ⎟ ⎝ ⎠
    47. 47. Find the terminal point if: 7π 55π 4. θ = − 5. θ = 4 6 8π 7π 55π 48π − − 4 4 6 6 π 7π same as 4 6 ⎛ 2 2 ⎞ ⎜ 2 , 2 ⎟ ⎝ ⎠
    48. 48. Find the terminal point if: 7π 55π 4. θ = − 5. θ = 4 6 8π 7π 55π 48π − − 4 4 6 6 π 7π same as 4 6 ⎛ 2 2 ⎞ ⎛ 3 1 ⎞ ⎜ 2 , 2 ⎟ ⎜ − 2 ,− 2 ⎟ ⎝ ⎠ ⎝ ⎠
    49. 49. HW #1Surround yourself with the best people you can find,delegate authority, and don’t interfere as long as thepolicy you’ve decided upon is being carried out. Ronald Reagan

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