Calculus with Parametric Curves: Horizontal and Vertical tangents?

Question

For this problem, we are asked to find the points on the curve where the tangent is horizontal or vertical.

1. x = Cos(3t), y = 2sin(t)


I know that dy/dt = 2Cos(t) and

dx/dt = -3sin(3t)


And that you set dy/dt equal to 0 to find the horizontal tangents, and dx/dt equal to 0 to find the vertical tangents. However, I am wondering what are the correct restrictions to apply in each case (for dx/dt and dy/dt) to find the correct number of the respective horizontal and vertical tangents. For instance, since dy/dt involves cosine, would I restrict it to values between 0 and pi (including these endpoints as well) and thus end up with a result that the horizontal tangent only occurs at pi/2, since this is the only location within the interval between 0 and pi where the cosine is 0?

Similarly, for dx/dt, since it involves sine, would I restrict it to 0 to pi/2 (including these endpoints as well) and thus say that the vertical tangents occurs at 0 and pi/3 (pi/3 because of the 3 in the expression, which cancels out with the 3 in pi/3 making it just pi, where sine is 0) because these are the only points in the given expression where the sine is 0 for the interval of 0 to pi/2?

Is all of this correct? If not, please indicate what are the correct restrictions to apply in each case (dx/dt and dy/dt), and why they are not what I listed above.

Answer 1

You've got the basic idea that this is a parametric curve so you are plotting y vs. x, not y vs. t or x vs. t. and so the tangents you want are dy/dx.

But that's where you start getting into trouble.

Most of the time, dy/dx is the same as (dy/dt)/(dx/dt), but that clearly isn't true when dx/dt is 0. At those points you may have a vertical tangent (if dy/dt were not 0) or you may have something else (if dy/dt were also 0)

Part of the interesting elements of this problem is that there will be some values of t for each case. Characterizing all the points is part of the problem.

So, what are the values of t at which both 2 cos(t) and -3 sin(3t) are zero? What is dy/dx then? (Clearly the tangent can't be both horizontal and vertical, but it can be something completely different :-)

What are the other points at which 2 cos(t) is 0? And the other points at which -3 sin(3t) is zero?

And since all of these are, as you've pointed out, periodic, the answers should be an equation in N for each answer.