Problem

Figure 3 shows a sketch of the curve $C$ with parametric equations

$$x=1+3\tan t\qquad y=2\cos 2t\qquad -\frac{\pi}{6}\le t\le \frac{\pi}{3}$$

The curve crosses the $x$-axis at point $P$, as shown in Figure 3.

(a) Find the equation of the tangent to $C$ at $P$, writing your answer in the form $y=mx+c$, where $m$ and $c$ are constants to be found.

The curve $C$ has equation $y=\mathrm{f}(x)$, where $\mathrm{f}$ is a function with domain $\left[k,1+3\sqrt{3}\right]$

(b) Find the exact value of the constant $k$.

(c) Find the range of $\mathrm{f}$.