# Help Polar, Horizontal Tangents.

## Homework Statement

For the three leaf rose: r = $$sin(3\theta)$$, find the co-ordinates of the five points where the tangent lines are horizontal. Interpret the significance of the five points.

## Homework Equations

$$\frac{dy}{dx} = \frac{r'sin(\theta)+ rcos(\theta)}{r'cos(\theta) - rsin(\theta)}$$

## The Attempt at a Solution

To find horizontal tangents we have to set the $$\frac{dy}{d\theta}$$ = 0 and solve. ( providing $$\frac{dx}{d\theta}$$ is not 0.

I found that $$\frac{dy}{d\theta}$$ = $$3cos(3\theta)sin(\theta) + sin(3\theta)cos(\theta) = 0$$ @ $$\theta)$$ = 0, pi, pi/2, -pi/2. I can't seem to find a fifth 0. for instance @ 2pi the tangent must be 0 as well, but i'm not sure.

When I graph sin(3$$\theta$$) using polarplot in maple. I see that clearly that they are four, one at the origin, one down to the bottom and two to the top.A fifth one is possible since the curve crosses the origin twice between 0..2pi. However none of my above points would plots the two points to the top of the curve. why? Maple gives me two solutions +-arctan(1/10*sqrt(6)*sqrt(10), which would make since since the two horizontal tangents @ the top are symmetric. But i'm not sure how they got to this +-arctan(1/10*sqrt(6)*sqrt(10).

My questions are: How do I find that fifth zero and why don't the zero's that I found represent the two tangents lines at the top of the curve? I'm guessing I didn't find all the zeros.
Thank you in advance.

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