The curve shown in Figure 3 has parametric equations $x = t - 4 \, ext{sin} \, t, \ y = 1 - 2 \, \text{cos} \, t, \quad \frac{2 \pi}{3} \leq t \leq \frac{2 \pi}{3}$ The point $A$, with coordinates $(k, 1)$, lies on the curve - Edexcel - A-Level Maths Pure - Question 8 - 2014 - Paper 8

To find the gradient of the curve at point $A$ where $t = \frac{3\pi}{2}$, we first calculate ( \frac{dx}{dt} ) and ( \frac{dy}{dt} ):

1. Derivative of $x$:

$\frac{dx}{dt} = 1 - 4 \text{cos} \, t$

2. Derivative of $y$:

$\frac{dy}{dt} = 2 \text{sin} \, t$

Next, we evaluate ( \frac{dx}{dt} ) and ( \frac{dy}{dt} ) at $t = \frac{3\pi}{2}$:

• For $\frac{dx}{dt}$: $\frac{dx}{dt} = 1 - 4 \text{cos} \left( \frac{3\pi}{2} \right) = 1 - 4(0) = 1$

• For $\frac{dy}{dt}$: $\frac{dy}{dt} = 2 \text{sin} \left( \frac{3\pi}{2} \right) = 2(-1) = -2$

Thus, the gradient ($m$) of the curve is given by:

$m = \frac{dy/dt}{dx/dt} = \frac{-2}{1} = -2$

To find the value of $t$ where the gradient is $-\frac{1}{2}$:

Given the equation for the gradient:

$\frac{dy}{dx} = \frac{2 \text{sin} \, t}{-4 + 4 \text{cos} \, t} = -\frac{1}{2}$

Cross-multiplying results in:

$2(2 \text{sin} \, t) = -(-4 + 4 \text{cos} \, t)$

This simplifies to:

$4 \text{sin} \, t = 4 - 4 \text{cos} \, t$

Dividing through by 4 gives:

$\text{sin} \, t = 1 - \text{cos} \, t$

Using the identity $ext{sin}^2 t + \text{cos}^2 t = 1$ and substituting $ext{sin} \, t = 1 - \text{cos} \, t$:

$(1 - \text{cos} \, t)^2 + \text{cos}^2 \, t = 1$

Expanding and simplifying:

$1 - 2\text{cos} \, t + \text{cos}^2 \, t + \text{cos}^2 \, t = 1$

This leads to:

$2 \text{cos}^2 \, t - 2\text{cos} \, t = 0$

Factoring out $2\text{cos} \, t$:

$2 \text{cos} \, t (\text{cos} \, t - 1) = 0$

So $\text{cos} \, t = 0$ or $\text{cos} \, t = 1$. The solution gives:

• When $\text{cos} \, t = 0, \text{sin} \, t = 1,\ t = \frac{\pi}{2} + 2n\pi \text{ and } t = \frac{3\pi}{2} + 2n\pi, n \in \mathbb{Z}$
• The relevant $t$ is $\frac{3\pi}{2} + 2n\pi$;

Next, consider the value of $t = \frac{3\pi}{2}$:

Consolidating the numerical value we find:

$\frac{3}{2}\pi = 4.7124$ (4 decimal places) Thus, the value of $t$ is approximately $4.7124$.