Inverse Trigonometric Functions

Inverse trigonometric functions are explored interactively using an applet. You may want to go through an interactive tutorial on the definition of the inverse function first.

The three trigonometric functions studied in this tutorial are: arcsin(x), arccos(x) and arctan(x).

The exploration is carried out by analyzing the graph of the function and the graph of its inverse. The domain and range of each of the above functions are also explored. Follow the steps in the tutorial below.
Interactive Tutorial

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click on the button above "click here to start" and MAXIMIZE the window obtained.

  1. Tutorial on arcsin(x)
    1. Select "inverse sine" in the left panel. The graph in blue is the graph of the restricted sine function defined by:
      
      
      
      f(x) = sin(x) where x is in [-pi/2 , pi/2]
      Check that the range of f(x) is [-1,1]. See that this function is a one-to-one function. In red is the arcsin(x) function, the inverse of f(x) defined above. Check that the domain of arcsin(x) is given by the interval [-1,1]. See that the range of arcsin(x) is given by the interval [-pi/2 , pi/2].

    2. Compare the domain of f(x) and the range of its inverse. Compare the range of f(x) and the domain of its inverse

    3. Use the slider to move a marker (blue) along the graph of f(x) and another marker (red) along the graph of its inverse. The coordinates of the two markers are shown above in blue and red. Compare the x-coordinate of the point on the graph of f(x) with the y-coordinate of the point on the graph of arcsin(x). Compare the y-coordinate of the point on the graph of f(x) with the x-coordinate of the point on the graph of arcsin(x). Compare the positions of the two points with respect to the line y = x (in green).

    4. Exercise: Use the applet to find an exact value to the following: arcsin(0), arcsin(-1/2), arcsin(-1).

  2. Tutorial on arccos(x)
    1. Select "inverse cosine" in the left panel. The graph in blue is the graph of the restricted cosine function defined by:
      g(x) = cos(x) where x is in [0 , pi]
      Check that the range of g(x) is [-1,1]. See that this function is a one-to-one function. In red is the arccos(x) function, the inverse of g(x) defined above. Check that the domain of arccos(x) is given by the interval [-1,1]. See that the range of arcsin(x) is given by the interval [0 , pi].

    2. Compare the domain of g(x) and the range of its inverse. Compare the range of g(x) and the domain of its inverse

    3. Use the slider to move a marker (blue) along the graph of g(x) and another marker (red) along the graph of its inverse. The coordinates of the two markers are shown above in blue and red. Compare the x-coordinate of the point on the graph of g(x) with the y-coordinate of the point on the graph of arccos(x). Compare the y-coordinate of the point on the graph of g(x) with the x-coordinate of the point on the graph of arccos(x). Compare the positions of the two points with respect to the line y = x (in green).

    4. Exercise: Use the applet to find an exact value to the following: arccos(0), arccos(1/2), arccos(1).

  3. Tutorial on arctan(x)
    1. Select "inverse tangent" in the left panel. The graph in blue is the graph of the restricted tangent function defined by:
      h(x) = tan(x) where x is in (-pi/2 , pi/2)
      Note the open interval above. tan(x) is undefined at -pi/2 and pi/2. Graphically it has vertical asymptotes (shown in broken lines) at these values of x . The range of h(x) is (-inf , +inf). ("inf" means infinity). See that this function is a one-to-one function. In red is the arctan(x) function, the inverse of h(x) defined above. see that the domain of arctan(x) is given by the interval (-inf,+inf). See that the range of arctan(x) is given by the interval (-pi/2 , pi/2). arctan(x) has horizontal asymptotes at y = -pi/2 and y = -pi/2.(shown in broken red lines)

    2. Compare the domain of h(x) and the range of its inverse. Compare the range of h(x) and the domain of its inverse

    3. Use the slider to move a marker (blue) along the graph of h(x) and another marker (red) along the graph of its inverse. The coordinates of the two markers are shown above in blue and red. Compare the x-coordinate of the point on the graph of h(x) with the y-coordinate of the point on the graph of arctan(x). Compare the y-coordinate of the point on the graph of h(x) with the x-coordinate of the point on the graph of arctan(x). Compare the positions of the two points with respect to the line y = x (in green).

    4. Exercise: Use the applet to find an exact value to the following: arctan(0), arctan(-1), arctan(1).

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    Inverse Trigonometric Functions