What is the domain for the principal branch of sinx?
By convention, the interval −π/2,π/2 is taken as the domain of the principal branch of sinx.
On this interval, sinx is strictly increasing, and hence, one-to-one, so it is invertible.
Figure A-7.7(a) Graph of the principal branch of sinx
The graph of fx is the set of all points x,fx. The graph of the inverse function f−1x, is the set of all points for which the ordinate and abscissa has been reversed, that is, the set of all points fx,x.
The inverse of fx can be graphed parametrically by having Maple graph the parametric representation fx,x.
Since the inverse of fx=sinx is the arcsine function, graphing parametrically arcsinx,x yields the graph of the inverse of the inverse, that is, the graph of sinx. This is obtained in Figure A-7.7(c) with the
applied to the list arcsinx,x, as per Figure A-7.7(b).
Figure A-7.7(b) Interactive Plot Builder applied to the list arcsinx,x
Figure A-7.7(c) Graph of the parametric representation arcsinx,x
Only that part of the sine curve coincident with its principal branch "survives" this graphing of the inverse of the inverse. Hence, it can be inferred from the graph that the domain of the principal branch is the interval −π/2,π/2.
Alternatively, use the Context Panel to launch the Plot Builder as per the following, selecting the option 2-D plot (parametric).
Obtain Figure A-7.7(c) by executing the command
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