SOLVING FOR THE ANGLE SUBTENDING ARC AND CHORD
Next we shall develop the Maclaurin Series for xy=sin y.  
Differentiating implicitly, where y=f(x) and y(0)=f(0),
The rest are graphs for the various values of n, which can go from n=1 to infinity.
Use the root to the cubic method for n=0, and use the Maclaurin solution for n=1 to
infinity.  The closer B/A gets to zero, the more solutions there are.  This calculation is useful
for multiple revolutions of arclength around the circle.  (While 0<B<2radius, 0<A<infinity.)  
The last illustration shows the cumulative representation of the curve in Quadrant I and II
only.
In the development below, use top-down substitution to arrive at
q/2=f(B/A).  Seven derivatives are calculated although more can be
taken, and are used in the Maclaurin expansion of
q/2.

These identities were arrived at by implicitly differentiating

xy=sin y seven times, then setting x=0 and solving for y', y'', y''' and
so on.
The above equations were obtained by setting x=0 in the equations
below, which can be used to derive further derivatives if desired:
Using the above prescribed method (top-down substitutions), the above
illustration shows its failure (red points) in representing the graph (the
black curve).  The values used are in the table at the lower right
corner.
© 2014 Jon L. Giffen

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In the equation, xy=sin y, when x=0 y cannot=0 since by
L'Hospital's rule (sin y)/y = 0/0 or cos y/1 = 1/1=1.  Therefore
y=n
p, where n=1,2,3,...  However this does determine that at y=0,
B/A=1

The equations below have not yet been purged of x=0 and can be
used to find further terms to the Maclaurin Series: