6 The polynomials \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined by $$\mathrm { f } ( x ) = 4 x ^ { 3 } + a x ^ { 2 } + 8 x + 15 \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 2 } + b x + 18$$ where \(a\) and \(b\) are constants.

1. Given that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
2. Given that the remainder is 40 when \(\mathrm { g } ( x )\) is divided by \(( x - 2 )\), find the value of \(b\).
3. When \(a\) and \(b\) have these values, factorise \(\mathrm { f } ( x ) - \mathrm { g } ( x )\) completely.
4. Hence solve the equation \(\mathrm { f } ( \operatorname { cosec } \theta ) - \mathrm { g } ( \operatorname { cosec } \theta ) = 0\) for \(0 < \theta < 2 \pi\).