Parametric question...confused on trig =/

**AshleyT** · December 31st 2008, 07:23 AM

Question:
A curve has parametric equations
x = 3 cos2 t, y = sin 2t, 0 ≤ t < π.
(i) Show that
= $\frac{dy}{dx} = -\frac{2}{3}cot 2t$
(ii) Find the coordinates of the points where the tangent to the curve is parallel to the x-axis

Ok, the first bit i did no problem, but its the second part because the gradient must be 0 correct?
meaning cot 2t = 0
However you cannot do the inverse of this because when cot t = 0...its an asymptote...

Thanks for any help

**Laurent** · December 31st 2008, 08:40 AM

Originally Posted by AshleyT

Question:
A curve has parametric equations
x = 3 cos2 t, y = sin 2t, 0 ≤ t < π.
(i) Show that
= $\frac{dy}{dx} = -\frac{2}{3}cot 2t$
(ii) Find the coordinates of the points where the tangent to the curve is parallel to the x-axis

Ok, the first bit i did no problem, but its the second part because the gradient must be 0 correct?
meaning cot 2t = 0
However you cannot do the inverse of this because when cot t = 0...its an asymptote...

Thanks for any help

For (i), I find $\frac{dy}{dx} = -\frac{1}{3}\cot 2t$ .

As for (ii), you must be confusing with something else: you indeed need to find points where $0=\frac{dy}{dx}$ , which means $0=\cot 2t=\frac{\cos 2t}{\sin 2t}$ . This is 0 when $\cos 2t=0$ . We have $0\leq 2t<2\pi$ , hence the solutions are $2t=\frac{\pi}{2}$ and $2t=\frac{3\pi}{2}$ . It remains to find $(x,y)$ for these $t$ .

**AshleyT** · December 31st 2008, 08:56 AM

Originally Posted by Laurent

For (i), I find $\frac{dy}{dx} = -\frac{1}{3}\cot 2t$ .

As for (ii), you must be confusing with something else: you indeed need to find points where $0=\frac{dy}{dx}$ , which means $0=\cot 2t=\frac{\cos 2t}{\sin 2t}$ . This is 0 when $\cos 2t=0$ . We have $0\leq 2t<2\pi$ , hence the solutions are $2t=\frac{\pi}{2}$ and $2t=\frac{3\pi}{2}$ . It remains to find $(x,y)$ for these $t$ .

Thanks! I think i just managed to confuse myself.

For part one though: How?
Cuz you divide
2cos 2t by -3sin 2t
and get -2/3 cot 2t