Figure 2 shows a sketch of the curve C with parametric equations $x = 4 ext{sin}igg(t + rac{ ext{π}}{6}igg),$ $y = 3 ext{cos}(2t), ext{ } 0 ext{ } extless ext{ } t ext{ } extless ext{ } 2 ext{π}$ (a) Find an expression for $\frac{dy}{dx}$ in terms of $t$ - Edexcel - A-Level Maths Pure - Question 6 - 2012 - Paper 8

For $\frac{dy}{dx} = 0$, we set the numerator of our earlier expression to zero:

$-6\text{sin}(2t) = 0$

This gives:

$\text{sin}(2t) = 0$

The solutions for $2t = n\pi$ where $n$ is an integer leads to:

$t = \frac{n\pi}{2}$

Considering the interval $0 < t < 2\pi$, we find:

• For $n = 0$, $t = 0$ (out of range)
• For $n = 1$, $t = \frac{\pi}{2}$
• For $n = 2$, $t = \pi$
• For $n = 3$, $t = \frac{3\pi}{2}$
• For $n = 4$, $t = 2\pi$ (out of range)

So the valid $t$ values are: $t = \frac{\pi}{2}, \pi, \frac{3\pi}{2}$

Next, we calculate the corresponding $x$ and $y$ coordinates:

1. For $t = \frac{\pi}{2}$:

   • $x = 4\text{sin}\bigg(\frac{\pi}{2} + \frac{\pi}{6}\bigg) = 4\text{sin}\bigg(\frac{2\pi}{3}\bigg) = 4(\frac{\sqrt{3}}{2}) = 2\sqrt{3}$
   • $y = 3\text{cos}(\pi) = -3$
   • Coordinates: $(2\sqrt{3}, -3)$

2. For $t = \pi$:

   • $x = 4\text{sin}\bigg(\pi + \frac{\pi}{6}\bigg) = 4\text{sin}\bigg(\frac{7\pi}{6}\bigg) = -4(\frac{1}{2}) = -2$
   • $y = 3\text{cos}(2\pi) = 3$
   • Coordinates: $(-2, 3)$

3. For $t = \frac{3\pi}{2}$:

   • $x = 4\text{sin}\bigg(\frac{3\pi}{2} + \frac{\pi}{6}\bigg) = 4\text{sin}\bigg(\frac{10\pi}{6}\bigg) = 4\text{sin}\bigg(\frac{5\pi}{3}\bigg) = 4(-\frac{\sqrt{3}}{2}) = -2\sqrt{3}$
   • $y = 3\text{cos}(3\pi) = -3$
   • Coordinates: $(-2\sqrt{3}, -3)$

Thus, the coordinates where $\frac{dy}{dx} = 0$ are:

• $(2\sqrt{3}, -3)$
• $(-2, 3)$
• $(-2\sqrt{3}, -3)$