Answers to Tutorials on Unit Circle

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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