Intearctive Inverse Trigonometric Functions

Inverse trigonometric functions are explored interactively using the graphs below. 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

Sine and Arcsine Functions

Function point (blue): (0, 0)
Inverse point (red): (0, 0)
Relationship: \(f(0) = 0\), \(f^{-1}(0) = 0\)

Select Function

Move Point Along Function

x-value: 0.00

Properties

Sine Function: \(f(x) = \sin(x)\)
Restricted Domain: \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)
Range: \([-1, 1]\)
Arcsine Function: \(f^{-1}(x) = \arcsin(x)\)
Domain: \([-1, 1]\)
Range: \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)

Exercise

Find exact values: \(\arcsin(0)\), \(\arcsin(1)\), \(\arcsin(-1)\)

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:
    \(f(x) = \sin(x) \quad \text{where} \quad x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)
    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 \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\).
  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 app above to find an exact value to the following: \(\arcsin(0)\), \(\arcsin(1)\), \(\arcsin(-1)\).

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:
    \(g(x) = \cos(x) \quad \text{where} \quad x \in [0, \pi]\)
    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 \(\arccos(x)\) 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 app above to find an exact value to the following: \(\arccos(0)\), \(\arccos(-1)\), \(\arccos(1)\).

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:
    \(h(x) = \tan(x) \quad \text{where} \quad x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

    Note the open interval above. \(\tan(x)\) is undefined at \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). Graphically it has vertical asymptotes (shown in broken lines) at these values of \(x\).

    The range of \(h(x)\) is \((-\infty, +\infty)\). 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 \((-\infty, +\infty)\). See that the range of \(\arctan(x)\) is given by the interval \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\). \(\arctan(x)\) has horizontal asymptotes at \(y = -\frac{\pi}{2}\) and \(y = \frac{\pi}{2}\) (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 app above to find an exact value to the following: \(\arctan(0)\), \(\arctan(-1)\), \(\arctan(1)\).

More References and Links to Inverse Trigonometric Functions