7 The equation of a curve, in polar coordinates, is
$$r = \sqrt { 3 } + \tan \theta , \quad \text { for } - \frac { 1 } { 3 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi$$
- Find the equation of the tangent at the pole.
- State the greatest value of \(r\) and the corresponding value of \(\theta\).
- Sketch the curve.
- Find the exact area of the region enclosed by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 4 } \pi\).