The parametric equations of a curve $C$ are $x=1-\sin t$ and $y=\cos t$, $0\le t\le 2\pi $.

Question

The parametric equations of a curve $C$ are $x=1-\sin t$ and $y=\cos t$, $0\le t\le 2\pi $. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.[1] Find the exact value(s) of $t$ at the point when the tangent is parallel to the $y$-axis.[2] Show that the equation of the normal at the point with parameter $p$ is $y=-x\cot p+\cot p$.[2] Show that the normal at the point $A$ with coordinates $\left( 1-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}} \right)$ cuts the curve again at another point $B$. Find the coordinates of $B$.[2]

Accepted Answer

(i) $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 	an t$ (ii) $t = \dfrac{\pi}{2}$ or $t = \dfrac{3\pi}{2}$ (iv) $B\left(1 + \dfrac{1}{\sqrt{2}}, -\dfrac{1}{\sqrt{2}}ight)$

ACJC Tutorial 13 Q1

ACJC Tutorial 13 Q1

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