User:Gauge00/aaa

Machin-like formula


 * $$ \mathrm{C}_8 \mathrm{H}_{18} + 12.5 \mathrm{O}_2

=> 8\mathrm{CO}_2 + 9\mathrm{H}_2\mathrm{O} $$

\arctan\frac{1}{y} = \frac{1}{y} - \frac{1}{3} \frac{1}{y^3} + \frac{1}{5} \frac{1}{y^5} - \frac{1}{7} \frac{1}{y^5} $$
 * $$ \frac{\pi}{4} = \arctan\frac{1}{2} + \arctan\frac{1}{5} + \arctan\frac{1}{8}

\ \ \ \text{(Strassnitzky's formula)} $$
 * here, $$ \arctan\frac{1}{5} + \arctan\frac{1}{8} = \arctan\frac{1}{3} $$


 * $$ \frac{\pi}{4} = 6\arctan\frac{1}{8} + 2\arctan\frac{1}{57} + \arctan\frac{1}{239} $$
 * $$ \frac{\pi}{4} = 4\arctan\frac{1}{5} - \arctan\frac{1}{70} + \arctan\frac{1}{99} $$


 * $$ \arctan\frac{1}{y} = \frac{1}{y} - \frac{1}{3y^3} + \frac{1}{5y^5} -\frac{1}{7y^5} +...$$
 * $$ = \frac{1}{y}\left( 1 - \frac{1}{3y^2} + \frac{1}{5y^4} -\frac{1}{7y^6} +... \right)

= \frac{1}{y}\left( 1 - \frac{1}{3y^2}\left(1 - \frac{3}{5y^2} + \frac{3}{7y^4} +... \right) \right)$$


 * $$ = \frac{1}{y} \left[ 1 +  \frac{1}{3y^2}\left(1 - \frac{3}{5y^2} + \frac{3}{7y^4} +...  \right)    \right] ^{-1}  =  \left[ y +  \frac{1}{3y}\left(1 - \frac{3}{5y^2} + \frac{3}{7y^4} +...  \right)    \right] ^{-1}  $$


 * $$ =  \left[ y +  \frac{1}{3y}\left[1 + \frac{3}{5y^2} - \frac{3}{7y^4} +...  \right]^{-1}    \right] ^{-1}

= \left[ y +  \left[3y + \frac{3^2}{5y} - \frac{3^2}{7y^3} +... \right]^{-1}   \right] ^{-1}$$


 * $$ \tan 2\theta = \frac{2 \tan \theta} {1 - \tan^2 \theta} $$
 * $$\tan 3\theta = \frac{3 \tan\theta - \tan^3\theta}{1 - 3 \tan^2\theta} $$
 * $$ 2\arctan\frac{1}{x} = \arctan\frac{2x}{x^2-1} $$
 * $$ 2\arctan\frac{1}{2x} = \arctan\frac{4x}{4x^2-1} = \arctan\frac{A(x) + x}{A(x)x - 1}$$


 * $$ 4x^2A(x) - 4x = 4x^2A(x)+ 4x^3 - A(x) -x $$
 * $$ A(x) = 3x + 4x^3 $$

Now,
 * $$ \frac{3^2}{5y} - \frac{3^2}{7y^3} + \frac{3^2}{9y^5} + ...

= \frac{3^2}{5y} \left( 1 - \frac{1}{7y^2} + \frac{1}{9y^4} + ... \right) = \frac{3^2}{5y} \left[ 1 + \left( \frac{1}{7y^2} - \frac{1}{9y^4} + ... \right) \right]^{-1} $$