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4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
4.6 graphs of trig functions and inverse trig functions
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4.6 graphs of trig functions and inverse trig functions

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  • 1. Graphs of Trig. Functions
  • 2. Graphs of Trig. Functions
    The graph of y=sin(x)
  • 3. Graphs of Trig. Functions
    The graph of y=sin(x)
  • 4. Graphs of Trig. Functions
    The graph of y=sin(x)
  • 5. Graphs of Trig. Functions
    The graph of y=sin(x)
  • 6. Graphs of Trig. Functions
    The graph of y=sin(x)
  • 7. Graphs of Trig. Functions
    The graph of y=sin(x)
  • 8. Graphs of Trig. Functions
    The graph of y=sin(x)
  • 9. Graphs of Trig. Functions
    The graph of y=sin(x)
  • 10. Graphs of Trig. Functions
    The graph of y=cos(x)
  • 11. Graphs of Trig. Functions
    The graph of y=cos(x)
  • 12. Graphs of Trig. Functions
    The graph of y=cos(x)
    90o
    180o
    0o
  • 13. Graphs of Trig. Functions
    The graph of y=cos(x)
    90o
    180o
    270o
    360o
    0o
  • 14. Graphs of Trig. Functions
    The graph of y=cos(x)
  • 15. Graphs of Trig. Functions
    The graph of y=cos(x)
    The graph of y=sin(x)
  • 16. Periodic Functions
  • 17. Periodic Functions
    Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that
    f(x) = f(x+b) for all x.
  • 18. Periodic Functions
    Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that
    f(x) = f(x+b) for all x.
    The graph of a periodic function:
     Frank Ma
    2006
    p
  • 19. Periodic Functions
    Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that
    f(x) = f(x+b) for all x.
    The smallest number p>0 such that f(x) = f(x+p)
    is called the period of f(x).
    The graph of a periodic function:
     Frank Ma
    2006
    p
  • 20. Periodic Functions
    Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that
    f(x) = f(x+b) for all x.
    The smallest number p>0 such that f(x) = f(x+p)
    is called the period of f(x).
    The graph of a periodic function:
     Frank Ma
    2006
    p
    one period
  • 21. Periodic Functions
    Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that
    f(x) = f(x+b) for all x.
    The smallest number p>0 such that f(x) = f(x+p)
    is called the period of f(x).
    Over every interval of length p, the graph of a periodic function repeats itself.
    The graph of a periodic function:
     Frank Ma
    2006
    p
    one period
  • 22. Periodic Functions
    Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that
    f(x) = f(x+b) for all x.
    The smallest number p>0 such that f(x) = f(x+p)
    is called the period of f(x).
    Over every interval of length p, the graph of a periodic function repeats itself.
    The graph of a periodic function:
     Frank Ma
    2006
    p
    x
    x+p
    p
    one period
  • 23. Periodic Functions
    Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that
    f(x) = f(x+b) for all x.
    The smallest number p>0 such that f(x) = f(x+p)
    is called the period of f(x).
    Over every interval of length p, the graph of a periodic function repeats itself.
    The graph of a periodic function:
     Frank Ma
    2006
    p
    x
    x+p
    p
    one period
    f(x) = f(x+p) for all x
  • 24. Periodic Functions
    sin(x) and cos(x) are periodic with period p=2π.
  • 25. Periodic Functions
    sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
    For y=cos(x):
  • 26. Periodic Functions
    sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
    For y=cos(x):
  • 27. Periodic Functions
    sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
    For y=cos(x):
  • 28. Periodic Functions
    sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
    For y=cos(x):
  • 29. Periodic Functions
    sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
    For y=cos(x):
  • 30. Periodic Functions
    sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
    For y=sin(x):
    0
  • 31. Periodic Functions
    sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
    For y=sin(x):
    0
  • 32. Periodic Functions
    sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
    For y=sin(x):
    0
  • 33. Periodic Functions
    sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
    For y=sin(x):
    0
  • 34. Periodic Functions
    The basic period for:
    y=sin(x)
  • 35. Periodic Functions
    The basic period for:
    y=sin(x) y=cos(x)
    1
    -1
  • 36. Periodic Functions
    The basic period for:
    y=sin(x) y=cos(x)
    1
    -1
    The Graph of Tangent
  • 37. Periodic Functions
    The basic period for:
    y=sin(x) y=cos(x)
    1
    -1
    The Graph of Tangent
    The function tan(x) is not defined when cos(x) is 0,
    i.e. when x = ±π/2, ±3π/2, ±5π/2, ..
  • 38. Periodic Functions
    The basic period for:
    y=sin(x) y=cos(x)
    1
    -1
    The Graph of Tangent
    The function tan(x) is not defined when cos(x) is 0,
    i.e. when x = ±π/2, ±3π/2, ±5π/2, ..
     Frank Ma
    2006
    As the values of x goes from 0 to π/2 the values of sin(x) goes from 0 to 1,
  • 39. Periodic Functions
    The basic period for:
    y=sin(x) y=cos(x)
    1
    -1
    The Graph of Tangent
    The function tan(x) is not defined when cos(x) is 0,
    i.e. when x = ±π/2, ±3π/2, ±5π/2, ..
     Frank Ma
    2006
    As the values of x goes from 0 to π/2 the values of sin(x) goes from 0 to 1, but the values of cos(x) goes from 1 to 0. So tan(x) goes from 0 to +∞.
  • 40. The Graph of Tangent
    Since tan(x) is odd, so as the values of x goes from
    0 to -π/2 we get the corresonding negative outputs
    for tan(x).
  • 41. The Graph of Tangent
    Since tan(x) is odd, so as the values of x goes from
    0 to -π/2 we get the corresonding negative outputs
    for tan(x). Specifically,
    x
    π/6
    0
    π/4
    π/3
    -π/2
    -π/6
    -π/4
    -π/3
    π/2
    tan(x)
  • 42. The Graph of Tangent
    Since tan(x) is odd, so as the values of x goes from
    0 to -π/2 we get the corresonding negative outputs
    for tan(x). Specifically,
    x
    π/6
    0
    π/4
    π/3
    -π/2
    -π/6
    -π/4
    -π/3
    π/2

    0
    1/3
    1
    3
    tan(x)
  • 43. The Graph of Tangent
    Since tan(x) is odd, so as the values of x goes from
    0 to -π/2 we get the corresonding negative outputs
    for tan(x). Specifically,
    x
    π/6
    0
    π/4
    π/3
    -π/2
    -π/6
    -π/4
    -π/3
    π/2

    0
    1/3
    1
    3
    -1/3
    -1
    -3
    -∞
    tan(x)
  • 44. The Graph of Tangent
    Since tan(x) is odd, so as the values of x goes from
    0 to -π/2 we get the corresonding negative outputs
    for tan(x). Specifically,
    x
    π/6
    0
    π/4
    π/3
    -π/2
    -π/6
    -π/4
    -π/3
    π/2

    0
    1/3
    1
    3
    -1/3
    -1
    -3
    -∞
    tan(x)
    0
    π/2
    -π/2
  • 45. The Graph of Tangent
    Since tan(x) is odd, so as the values of x goes from
    0 to -π/2 we get the corresonding negative outputs
    for tan(x). Specifically,
    x
    π/6
    0
    π/4
    π/3
    -π/2
    -π/6
    -π/4
    -π/3
    π/2

    0
    1/3
    1
    3
    -1/3
    -1
    -3
    -∞
    tan(x)
    The same pattern repeats itself every πinterval.
    0
    π/2
    -π/2
  • 46. The Graph of Tangent
    Since tan(x) is odd, so as the values of x goes from
    0 to -π/2 we get the corresonding negative outputs
    for tan(x). Specifically,
    x
    π/6
    0
    π/4
    π/3
    -π/2
    -π/6
    -π/4
    -π/3
    π/2

    0
    1/3
    1
    3
    -1/3
    -1
    -3
    -∞
    tan(x)
    The same pattern repeats itself every πinterval.
    In other words,
    y = tan(x) is a periodic function with period πas shown in the graph.
    0
    π/2
    -π/2
  • 47. The Graph of Tangent
    Since tan(x) is odd, so as the values of x goes from
    0 to -π/2 we get the corresonding negative outputs
    for tan(x). Specifically,
    x
    π/6
    0
    π/4
    π/3
    -π/2
    -π/6
    -π/4
    -π/3
    π/2

    0
    1/3
    1
    3
    -1/3
    -1
    -3
    -∞
    tan(x)
    The same pattern repeats itself every πinterval.
    In other words,
    y = tan(x) is a periodic function with period πas shown in the graph.
    π

    0
    π/2
    -π/2
    3π/2
    -3π/2
    y = tan(x)
  • 48. The Graph of Inverse Trig-Functions
    Recalll that for y = cos-1(x), then 0 < y < π.
  • 49. The Graph of Inverse Trig-Functions
    Recalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner.
  • 50. The Graph of Inverse Trig-Functions
    Recalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner.
    (-1, π)
    (0, π/2)
    (1, 0)
    -1
    1
    The graph of y = cos-1(x)
  • 51. The Graph of Inverse Trig-Functions
    Recalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner.
    (-1, π)
    (0, π/2)
    (1, 0)
    -1
    1
    The graph of y = cos-1(x)
    Remark: The above graphs of y = sin-1(x) and
    y = cos-1(x) are the complete graphs (i.e. that's all there is).
  • 52. The Graph of Inverse Trig-Functions
    The domain of y = tan-1(x) is all real numbers and
    the output y is restricted to -π/2 < y < π.
  • 53. The Graph of Inverse Trig-Functions
    The domain of y = tan-1(x) is all real numbers and
    the output y is restricted to -π/2 < y < π.
    x

    0
    1/3
    1
    3
    -1/3
    -1
    -3
    -∞
    tan-1(x)
  • 54. The Graph of Inverse Trig-Functions
    The domain of y = tan-1(x) is all real numbers and
    the output y is restricted to -π/2 < y < π.
    x

    0
    1/3
    1
    3
    -1/3
    -1
    -3
    -∞
    tan-1(x)
    π/6
    0
    π/4
    π/3
    π/2
  • 55. The Graph of Inverse Trig-Functions
    The domain of y = tan-1(x) is all real numbers and
    the output y is restricted to -π/2 < y < π.
    x

    0
    1/3
    1
    3
    -1/3
    -1
    -3
    -∞
    tan-1(x)
    π/6
    0
    π/4
    π/3
    -π/2
    -π/6
    -π/4
    -π/3
    π/2
  • 56. The Graph of Inverse Trig-Functions
    The domain of y = tan-1(x) is all real numbers and
    the output y is restricted to -π/2 < y < π.
    x

    0
    1/3
    1
    3
    -1/3
    -1
    -3
    -∞
    tan-1(x)
    π/6
    0
    π/4
    π/3
    -π/2
    -π/6
    -π/4
    -π/3
    π/2
    Here is the graph of y = tan-1(x)
    y = π/2
    (1,π/4)
    (0,0)
    (-1,-π/4)
    y = -π/2
  • 57. The Graph of Inverse Trig-Functions
    The domain of y = tan-1(x) is all real numbers and
    the output y is restricted to -π/2 < y < π.
    x

    0
    1/3
    1
    3
    -1/3
    -1
    -3
    -∞
    tan-1(x)
    π/6
    0
    π/4
    π/3
    -π/2
    -π/6
    -π/4
    -π/3
    π/2
    Here is the graph of y = tan-1(x)
    y = π/2
    (1,π/4)
    (0,0)
    (-1,-π/4)
    y = -π/2
    Remark: y =tan-1(x) has two horizontal asymptoes.

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