it does not define a function.
However, redefine as
(see “A Sprint to FTC“)
and it simplifies (2) to
Since for (3) gives It means
It is true that
The set defines a function.
In fact, defines , the inverse function of
Let us now examine qualitatively.
It follows that
is an increase function on .
for is concave on . It is convex on
Fig. 1 illustrated qualitatively
To compute for any given , see “A Cautionary Tale of Compute Inverse Trigonometric Functions“, “A Mathematical Allegory“.
Exercise-1 Define , the inverse function of
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