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:

Clearly, , 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 .

As I plug multiples of 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 rather than simply .

Obviously, I am doing something wrong, but I don't know what. Isn't the idea to simply look at where ? What am I missing?

Thanks!

Re: Finding Vertical Tangent of Parametric Curve

Hello, joatmon!

Your algebra/trig is off . . .

We have: .

We want: .

Hence: .

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Re: Finding Vertical Tangent of Parametric Curve

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