## Inverse Trigonometric Functions

The applet below illustrates the relationship between a trigonometric function and its inverse. The first graph is f(x), the "inner" function, and the second graph is g(x), the "outer" function. The final graph shows the composition g(f(x). Note the following:

• The input of f (the red segment) is the same as the input of the composition.
• The output of f (the green segment) is the input of g.
• The output of g (the blue segment) is the output of the composition.
• You can move the input (the red dot) around with your mouse to see how changing the input of f will change the output of the composition.

If f and g are true inverses, then g(f(x))=x and f(g(x))=x.
Starting with
f(x)=arcsin(x) and g(x)=sin(x) in the applet below, we see that indeed, g(f(x))=sin(arcsin(x))=x.
Now interchange f and g in the applet by choosing the second example in the list at the top and clicking on the Load Example button. What is
arcsin(sin(x))? How do explain your result?

FunctionComposition applet written by David Eck (http://math.hws.edu/javamath/index.html)

Remember that arcsin(sin(x))=x only on the interval [-pi/2,pi/2]. So outside this interval, this identity does not hold. For example, arcsin(sin(pi))=arcsin(0)=0 and arcsin(sin(3*pi/2))=arcsin(-1)=-pi/2.

• Now look at the 3rd and 4th examples in the applet to see the graphs of let tan(arctan(x)) and arctan(tan(x)). Explain your observations.
• Finally, look at the 5th and 6th examples in the applet to see the graphs of let cos(arccos(x)) and arccos(cos(x)). Again, explain your observations.