Finding Vertical Tangent of Parametric Curve

I know from graphing this that there are 6 vertical tangents to the following curve, but I can only determine two of them:

$\displaystyle x = \cos(3\theta)$ $\displaystyle y = 3 sin \theta$

Clearly, $\displaystyle \frac{dx}{d\theta} = -3\sin(3\theta)$, so I have to find the points at which this value is equal to zero. As I see it, this happens at each multiple of $\displaystyle \pi$.

As I plug multiples of $\displaystyle \pi$ into the original equation, I get back either (1,0) or (-1,0), so I know that these points are have vertical tangents. But, I can see from the graph that there are vertical tangents at each multiple of $\displaystyle \frac {\pi}{3}$ rather than simply $\displaystyle \pi$.

Obviously, I am doing something wrong, but I don't know what. Isn't the idea to simply look at where $\displaystyle \frac{dx}{d\theta} = 0$? What am I missing?

Thanks!

Re: Finding Vertical Tangent of Parametric Curve

Hello, joatmon!

Your algebra/trig is off . . .

We have: .$\displaystyle \begin{Bmatrix}x &= &\cos(3\theta) \\ y &=& 3\sin\theta \end{Bmatrix}$

We want: .$\displaystyle \frac{dx}{d\theta} \,=\,0$

Hence: .$\displaystyle -3\sin(3\theta) \:=\:0 $

. . . . . . . . $\displaystyle \sin(3\theta) \:=\:0$

. . . . . . . . . . . $\displaystyle 3\theta \:=\:\pi n $

. . . . . . . . . . . . $\displaystyle \theta \:=\:\frac{\pi}{3}n$

Re: Finding Vertical Tangent of Parametric Curve

Thanks. I had a feeling that I was missing something obvious.