8 It is given that \(\theta = \tan ^ { - 1 } \left( \frac { 3 x } { 2 } \right)\).
- By writing \(\theta = \tan ^ { - 1 } \left( \frac { 3 x } { 2 } \right)\) as \(2 \tan \theta = 3 x\), use implicit differentiation to show that \(\frac { \mathrm { d } \theta } { \mathrm { d } x } = \frac { k } { 4 + 9 x ^ { 2 } }\), where \(k\) is an integer.
[0pt]
[3 marks] - Hence solve the differential equation
$$9 y \left( 4 + 9 x ^ { 2 } \right) \frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosec } 3 y$$
given that \(x = 0\) when \(y = \frac { \pi } { 3 }\). Give your answer in the form \(\mathrm { g } ( y ) = \mathrm { h } ( x )\).
[0pt]
[7 marks]