4
\includegraphics[max width=\textwidth, alt={}, center]{76371b0f-0145-4cc4-a147-27bcd749816a-2_339_1395_1089_374} The diagram shows a semicircle \(A C B\) with centre \(O\) and radius \(r\). The tangent at \(C\) meets \(A B\) produced at \(T\). The angle \(B O C\) is \(x\) radians. The area of the shaded region is equal to the area of the semicircle.

1. Show that \(x\) satisfies the equation $$\tan x = x + \pi$$
2. Use the iterative formula \(x _ { n + 1 } = \tan ^ { - 1 } \left( x _ { n } + \pi \right)\) to determine \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.