## Inverse Trigonometric Functions

*(please wait while the applet loads...)*

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.

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