I have few difficulties when drawing inverse trigonometric functions graph. I am unable to draw the graph of $[\sin^{-1}x]$; the [$\cdot$] (the brackets] denote the greatest integer function of $\sin^{-1}x$. I am confused whether to take integers on x axis or angles in radians on $x$-axis .
I would be grateful if someone could guide me in drawing the above function’s graph.
The greatest integer function returns the greatest integer less than or equal to its argument. In your case $[\sin^{-1}(x)]$, the G.I.N.T (greatest integer function) takes angles as input but the angles vary based on the value of $x$ which is a real value. So the x-axis is for real values between -1 and 1.
So for the input to your G.I.N.T, we have to consider the range of sine inverse as those values are inputted in the argument. Because we know $$\sin^{-1}:[-1,1]\to[\frac{-\pi}{2},\frac{\pi}{2}]$$ and $\frac{\pi}{2}>1$, we have to consider the following domains (values of $x$) separately: $$[-1,-\sin(1)),[-\sin(1),0),[0,\sin(1)),[\sin(1),1]$$ You can obtain these partitions yourself by trying to find where the argument to G.I.N.T becomes an integer and also keeping in mind the range of the sine inverse function.
In the first domain, $\sin^{-1}(x)\in[\frac{-\pi}{2}-1)$, so its G.I.N.T is -2 here.
In the second domain, $\sin^{-1}(x)\in[-1,0)$, so its G.I.N.T is -1 here.
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Assuming you can do the other 2, you get: