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Chapter 5
Trigonometric Functions of Real Numbers
Chapter 5
Trigonometric Functions of Real Numbers

          5.1   The Unit Circle
Chapter 5
Trigonometric 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.
Assumptions
Assumptions

1. Much of this chapter is review for you
Assumptions

1. Much of this chapter is review for you

2. You will ask questions about concepts
   you don’t understand or skills you
   can’t do or remember
Assumptions

1. Much of this chapter is review for you

2. You will ask questions about concepts
   you don’t understand or skills you
   can’t do or remember

3. You have your Unit Circle with you in
   in class and when doing homework
Two Approaches to Trigonometry
Two Approaches to Trigonometry

1. Unit Circle (Chapter 5)
Two Approaches to Trigonometry

1. Unit Circle (Chapter 5)

2. Right Triangle (Chapter 6)
Two Approaches to Trigonometry

1. Unit Circle (Chapter 5)

2. Right Triangle (Chapter 6)

we work with both as each has its strengths
Two Approaches to Trigonometry

1. 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
The Unit Circle is the circle of radius 1
        centered at the origin
The Unit Circle is the circle of radius 1
        centered at the origin



                                    2       2
                                   x + y =1
⎛ 6 7 ⎞
Is the point   ⎜ 6 , 6 ⎟   on the Unit Circle?
               ⎝       ⎠
⎛ 6 7 ⎞
Is the point   ⎜ 6 , 6 ⎟       on the Unit Circle?
               ⎝       ⎠

                    2        2
                  x + y =1
⎛ 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 ⎞
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
⎛ 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
⎛ 35 ⎞
Point P ⎜   , y⎟ is on the Unit Circle in quadrant IV.
        ⎝ 6    ⎠
                         Find y .
⎛ 35 ⎞
Point P ⎜   , y⎟ is on the Unit Circle in quadrant IV.
        ⎝ 6    ⎠
                         Find y .

       2
⎛ 35 ⎞   2
⎜ 6 ⎟ + y = 1
⎝    ⎠
⎛ 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
⎛ 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
⎛ 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
⎛ 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
⎛ 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
When the initial ray rotates through some
     angle, θ , its ending position is called
the terminal ray and the point of intersection
  of the terminal ray and the Unit Circle is
               the terminal point.
Key Terminal Points
  (Your Unit Circle)
Key Terminal Points
  (Your Unit Circle)
Key Terminal Points
                    (Your Unit Circle)




These values are generated by using the properties of
30-60-90 and 45-45-90 triangles. You did this in AAT.
Find the terminal point if:

        5π
 1. θ =
         4
Find the terminal point if:

        5π
 1. θ =
         4
     ⎛    2    2 ⎞
     ⎜ − 2 ,− 2 ⎟
     ⎝           ⎠
Find the terminal point if:

        5π
 1. θ =
         4
     ⎛    2    2 ⎞
     ⎜ − 2 ,− 2 ⎟
     ⎝           ⎠

           π
  2. θ = −
           6
Find the terminal point if:

        5π
 1. θ =
         4
     ⎛    2    2 ⎞
     ⎜ − 2 ,− 2 ⎟
     ⎝           ⎠

           π
  2. θ = −
           6
      12π π
           −
       6     6
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
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 ⎟
       ⎝        ⎠
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 ⎟
       ⎝        ⎠
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 ⎟
       ⎝        ⎠
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 ⎟
       ⎝        ⎠
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 ⎟
       ⎝        ⎠
Find the terminal point if:

            7π
   4. θ = −
             4
Find the terminal point if:

            7π
   4. θ = −
             4
       8π 7π
           −
        4     4
Find the terminal point if:

            7π
   4. θ = −
             4
       8π 7π
           −
        4     4
            π
    same as
            4
Find the terminal point if:

            7π
   4. θ = −
             4
       8π 7π
           −
        4     4
            π
    same as
            4
       ⎛ 2 2 ⎞
       ⎜ 2 , 2 ⎟
       ⎝       ⎠
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 ⎟
       ⎝       ⎠
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 ⎟
       ⎝       ⎠
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 ⎟
       ⎝       ⎠
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 ⎟
       ⎝       ⎠                ⎝          ⎠
HW #1

Surround yourself with the best people you can find,
delegate authority, and don’t interfere as long as the
policy you’ve decided upon is being carried out.
                                        Ronald Reagan

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0501 ch 5 day 1

  • 2. Chapter 5 Trigonometric Functions of Real Numbers 5.1 The Unit Circle
  • 3. Chapter 5 Trigonometric 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.
  • 5. Assumptions 1. Much of this chapter is review for you
  • 6. Assumptions 1. Much of this chapter is review for you 2. You will ask questions about concepts you don’t understand or skills you can’t do or remember
  • 7. Assumptions 1. Much of this chapter is review for you 2. You will ask questions about concepts you don’t understand or skills you can’t do or remember 3. You have your Unit Circle with you in in class and when doing homework
  • 8. Two Approaches to Trigonometry
  • 9. Two Approaches to Trigonometry 1. Unit Circle (Chapter 5)
  • 10. Two Approaches to Trigonometry 1. Unit Circle (Chapter 5) 2. Right Triangle (Chapter 6)
  • 11. Two Approaches to Trigonometry 1. Unit Circle (Chapter 5) 2. Right Triangle (Chapter 6) we work with both as each has its strengths
  • 12. Two Approaches to Trigonometry 1. 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. The Unit Circle is the circle of radius 1 centered at the origin
  • 14. The Unit Circle is the circle of radius 1 centered at the origin 2 2 x + y =1
  • 15. ⎛ 6 7 ⎞ Is the point ⎜ 6 , 6 ⎟ on the Unit Circle? ⎝ ⎠
  • 16. ⎛ 6 7 ⎞ Is the point ⎜ 6 , 6 ⎟ on the Unit Circle? ⎝ ⎠ 2 2 x + y =1
  • 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. ⎛ 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. ⎛ 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. ⎛ 35 ⎞ Point P ⎜ , y⎟ is on the Unit Circle in quadrant IV. ⎝ 6 ⎠ Find y .
  • 21. ⎛ 35 ⎞ Point P ⎜ , y⎟ is on the Unit Circle in quadrant IV. ⎝ 6 ⎠ Find y . 2 ⎛ 35 ⎞ 2 ⎜ 6 ⎟ + y = 1 ⎝ ⎠
  • 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. ⎛ 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. ⎛ 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. ⎛ 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. ⎛ 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. When the initial ray rotates through some angle, θ , its ending position is called the terminal ray and the point of intersection of the terminal ray and the Unit Circle is the terminal point.
  • 28. Key Terminal Points (Your Unit Circle)
  • 29. Key Terminal Points (Your Unit Circle)
  • 30. Key Terminal Points (Your Unit Circle) These values are generated by using the properties of 30-60-90 and 45-45-90 triangles. You did this in AAT.
  • 31. Find the terminal point if: 5π 1. θ = 4
  • 32. Find the terminal point if: 5π 1. θ = 4 ⎛ 2 2 ⎞ ⎜ − 2 ,− 2 ⎟ ⎝ ⎠
  • 33. Find the terminal point if: 5π 1. θ = 4 ⎛ 2 2 ⎞ ⎜ − 2 ,− 2 ⎟ ⎝ ⎠ π 2. θ = − 6
  • 34. Find the terminal point if: 5π 1. θ = 4 ⎛ 2 2 ⎞ ⎜ − 2 ,− 2 ⎟ ⎝ ⎠ π 2. θ = − 6 12π π − 6 6
  • 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. 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. 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. 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. 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. 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. Find the terminal point if: 7π 4. θ = − 4
  • 42. Find the terminal point if: 7π 4. θ = − 4 8π 7π − 4 4
  • 43. Find the terminal point if: 7π 4. θ = − 4 8π 7π − 4 4 π same as 4
  • 44. Find the terminal point if: 7π 4. θ = − 4 8π 7π − 4 4 π same as 4 ⎛ 2 2 ⎞ ⎜ 2 , 2 ⎟ ⎝ ⎠
  • 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. 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. 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. 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. HW #1 Surround yourself with the best people you can find, delegate authority, and don’t interfere as long as the policy you’ve decided upon is being carried out. Ronald Reagan

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