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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:
If f and g are true inverses, then g(f(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.
Math 141-142 home page