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
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
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
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=np, where n=1,2,3,... However this does determine that at y=0,
The equations below have not yet been purged of x=0 and can be
used to find further terms to the Maclaurin Series: