Answers to Tutorials on Unit Circle

The answers to the questions in the tutorial in Unit Circle and Trigonometric Functions sin(x), cos(x), tan(x) are presented.


  1. Question:
    Is there a point P(x) that cannot have any values for its x or y-coordinates? The x and y-coordinates are cos(x) and sin(x), what is the domain of sin(x), what is the domain of cos(x)?

    Answer:
    No, any point P(x) on the unit circle have an x and a y coordinates. The domain of sin(x) is given by the interval (-infinity , +infinity). The domain of cos(x) is also given by the interval (-infinity , +infinity).

  2. Question:
    Explore the x-intercepts, the maximums and minimums (if any) of the graphs of sin(x) and cos(x) using the unit circle.

    Answer:
    In what follows, k is an integer.
    x-intercepts for sin(x) : for all values of x such that x = k*Pi
    x-intercepts for cos(x) : for all values of x such that x = Pi/2 + k*Pi
    maximums for sin(x) : for all values of x such that x = Pi/2 + k*(2*Pi)
    maximums for cos(x) : for all values of x such that x = k*(2*Pi)
    minimums for sin(x) : for all values of x such that x = 3*Pi/2 + k*(2*Pi)
    minimums for cos(x) : for all values of x such that x = Pi + k*(2*Pi)

  3. Question: Using the unit circle, do you think that any of the coordinates of a point on the circle can be larger than 1 or smaller than -1. Why do you think that sin(x) and cos(x) cannot be larger than 1 or smaller than -1?

    Answer:
    Since the point is on the unit circel (radius = 1), its x and y coordinates cannot be larger than 1 or smaller that -1. cos(x) is the x coordinate and sin(x) is the y coordinate and cannot be larger than 1 or smaller than -1. This explain the range of sin(x) and cos(x).
    -1 <= cos(x) >= 1
    -1 <= sin(x) >= 1

  4. Question:
    Explore the periodicity of sin(x), cos(x) and tan(x).

    Answer:
    If point p(x) is moved around the circle (x changes) we can see that a after a full perimeter (2*Pi) the values of sin(x) and cox(x) are repeated. Hence the period of sin(x) and cos(x) is 2*Pi. The period of tan(x) is Pi.

  5. Question:
    tan(x)is the ratio y-coordinate / x-coordinate and whenever the x-coordinate of point P(x) is equal to zero, we cannot define tan(x).Find these points for x between zero and 2*Pi. What is the domain of tan(x)?

    Answer:
    The x coordinate is equal to zero for points P(x) such that x = Pi/2 and x = 3*Pi/2 and therefore the tangent is not defined at these values.

    The domain of tan(x) will include all real numbers except all values of x such that x = Pi/2 + k*Pi

  6. Question:
    At these same points where tan(x)is undefined, the graph of tan(x) on the right shows an asymptotic behavior, Explain. What do you think is the range of tan(x)? Find all x-intercepts of tan(x) between 0 and 2Pi (inclusive). Explain their positions on the x-axis.

    Answer:
    The asymptotic behavior of the graph of tan(x) at x = Pi/2 + k*Pi indicates that as x approaches Pi/2 + k*Pi, the value of tan(x) either becomes smaller without bound or larger without bound. This leads to saying that the
    range of tan(x) is given by the interval (-infinity , +infinity).
    The x intercepts of the graph of tan(x) between o and 2Pi are at (0,0) , (Pi,0) and (2Pi , 0). The positions of all the x intercepts on the x axis are given by (k*Pi, 0).





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