Help Polar, Horizontal Tangents.

[Fresh(2^)]

Homework Statement

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

Homework Equations

[tex]\frac{dy}{dx} = \frac{r'sin(\theta)+ rcos(\theta)}{r'cos(\theta) - rsin(\theta)}[/tex]

The Attempt at a Solution

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

I found that [tex]\frac{dy}{d\theta}[/tex] = [tex]3cos(3\theta)sin(\theta) + sin(3\theta)cos(\theta) = 0[/tex] @ [tex]\theta)[/tex] = 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[tex]\theta[/tex]) 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.

 

[mighty2000]

it is important to understand the significance of the five points where the tangent lines are horizontal in the three leaf rose. These points represent critical points on the curve where the slope of the tangent line is equal to 0. This means that at these points, the curve changes direction from increasing to decreasing or vice versa.

In this case, the five points represent the "petals" of the three leaf rose, where the curve changes direction from increasing to decreasing or vice versa. The first four points, at 0, π, π/2, and -π/2, correspond to the four outer petals of the rose. The fifth point, at 2π, corresponds to the inner petal at the origin.

The significance of these points lies in the fact that they represent the points of symmetry on the curve. The three leaf rose is a symmetrical shape, and these points of horizontal tangents represent the points of symmetry where the curve is mirrored.

To find the fifth point, you can use the equation \frac{dy}{dx} = 0 and solve for θ. Alternatively, you can use the equation r = sin(3θ) to plot the curve and visually identify the fifth point.

In terms of the solution given by Maple, it is important to understand that trigonometric functions have multiple solutions. In this case, the equation \frac{dy}{dθ} = 0 has multiple solutions, and the solution given by Maple, ±arctan(1/10√6√10), represents the two possible solutions at the fifth point.