6 In polar coordinates, the equation of a curve, \(C\), is \(r = 6 \sin ( 2 \theta ) \sinh \left( \frac { 1 } { 3 } \theta \right)\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
The pole of the polar coordinate system corresponds to the origin of the cartesian system and the initial line corresponds to the positive \(x\)-axis.

1. Explain how you can tell that \(C\) comprises a single loop in the first quadrant, passing through the pole. The incomplete table below shows values of \(r\) for various values of \(\theta\).

   \(\theta\)	0	\(\frac { 1 } { 12 } \pi\)	\(\frac { 1 } { 6 } \pi\)	\(\frac { 1 } { 4 } \pi\)	\(\frac { 1 } { 3 } \pi\)	\(\frac { 5 } { 12 } \pi\)	\(\frac { 1 } { 2 } \pi\)
   \(r\)	0	0.262			1.851

2. Use the copy of the table and the polar coordinate system diagram given in the Printed Answer Booklet to complete the table and sketch \(C\). The point on \(C\) which is furthest away from the pole is denoted by \(A\) and the value of \(\theta\) at \(A\) is denoted by \(\phi\).
3. Show that \(\phi\) satisfies the equation \(\phi = \frac { 3 } { 2 } \ln \left( \frac { 6 - \tan 2 \phi } { 6 + \tan 2 \phi } \right)\)
4. You are given that the relevant solution of the equation given in part (c) is \(\phi = 1.0207\) correct to 5 significant figures. Find the distance from \(A\) to the pole. Give your answer correct to \(\mathbf { 3 }\) significant figures.