Problem

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve $C$ has parametric equations

$$x=\sin t-3\cos^2 t\qquad y=3\sin t+2\cos t\qquad 0\le t\le 5$$

(a) Show that

$$\frac{dy}{dx}=3$$

where $t=\pi$

The point $P$ lies on $C$ where $t=\pi$

(b) Find the equation of the tangent to the curve at $P$ in the form $y=mx+c$ where $m$ and $c$ are constants to be found.

Given that the tangent to the curve at $P$ cuts $C$ at the point $Q$

(c) show that the value of $t$ at point $Q$ satisfies the equation

$$9\cos^2 t+2\cos t-7=0$$

(d) Hence find the exact value of the $y$ coordinate of $Q$