By sketching a suitable pair of graphs, show that the equation
$$\operatorname { cosec } x = \frac { 1 } { 2 } x + 1$$
where \(x\) is in radians, has a root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
Verify, by calculation, that this root lies between 0.5 and 1 .
Show that this root also satisfies the equation
$$x = \sin ^ { - 1 } \left( \frac { 2 } { x + 2 } \right)$$
Use the iterative formula
$$x _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 2 } { x _ { n } + 2 } \right)$$
with initial value \(x _ { 1 } = 0.75\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.