8.
$$f ( x ) = \frac { 3 } { 13 + 6 \sin x - 5 \cos x }$$
Using the substitution \(t = \tan \left( \frac { x } { 2 } \right)\)
- show that \(\mathrm { f } ( x )\) can be written in the form
$$\frac { 3 \left( 1 + t ^ { 2 } \right) } { 2 ( 3 t + 1 ) ^ { 2 } + 6 }$$
- Hence solve, for \(0 < x < 2 \pi\), the equation
$$\mathrm { f } ( x ) = \frac { 3 } { 7 }$$
giving your answers to 2 decimal places where appropriate.
- Use the result of part (a) to show that
$$\int _ { \frac { \pi } { 3 } } ^ { \frac { 4 \pi } { 3 } } f ( x ) d x = K \left( \arctan \left( \frac { \sqrt { 3 } - 9 } { 3 } \right) - \arctan \left( \frac { \sqrt { 3 } + 3 } { 3 } \right) + \pi \right)$$
where \(K\) is a constant to be determined.