Solution of Arcsin Series

y = arcsin x therefore x = sin y dy/dx = 1 / [ dx/dy ] Substituting : d(arcsin x)/dx = 1 / [ d(sin y)/dy ] = 1/cos y But cos 2 y = 1 – sin 2 y and sin y = x Substituting : cos 2 y = 1 – x 2

Expanding the right hand term (Binomial Theorem)

Integrating both sides of the equation

Evaluation of π

sin π/6 = 1/2 arcsin ( π/6 ) = 1/2 + [ 1 × (1/2) 3 ] / [ 2 × 3 ]                     + [ 1 × 3 × (1/2) 5 ] / [ 2 × 4 × 5 ]                     + [ 1 × 3 × 5 × (1/2) 7 ] / [ 2 × 4 × 6 × 7 ] + ...                     = 1/2 + 1/48 + 3/1280 + 15/43008 + 105/1769472 ...                     = .52359 ... π = 6 × .52359 ... = 3.1415 ... continue adding terms for accuracy