(41C) Method of Successive Substitutions

[Albert Chan]

(06-10-2022 07:20 AM)Thomas Klemm Wrote:  The other example is a bit more complicated since ASIN returns a value that doesn't come close to a solution.
So we need to adjust this by adding \( 2 \pi \):

\(
\begin{align}
f(x) &= \frac{\pi}{\sin(x)} \\
\\
f^{-1}(x) &= \sin^{-1}\left(\frac{\pi}{x}\right) + 2 \pi \\
\end{align}
\)

It may be better to rearrange it so we could visualize the curves.

x = pi/sin(x)
sin(x) = pi/x

RHS ≥ 0    → LHS: x (mod 2*pi) = 0 .. pi
LHS ≤ 1    → RHS: x ≥ pi

First positive solution: x = 2*pi .. 3*pi

>>> x = 6.28
>>> for i in range(10): x += pi/x - sin(x); print x
...
6.78343890905
6.76691767331
6.76608868702
6.76605061437
6.76604887382
6.76604879426
6.76604879063
6.76604879046
6.76604879045
6.76604879045