Horizontal and Vertical Tangents for Polar Equation

In summary, the problem requires finding points where the polar equation r2 = 4cos(2θ) has horizontal and vertical tangents. The attempt at solving the problem involves using implicit differentiation to find r'(θ), which leads to an equation involving both r and θ. This can be solved with the original equation for the curve, resulting in two equations and two unknowns.
  • #1
EnlightenedOne
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Homework Statement


Find the points at which the following polar equation has horizontal and vertical tangents:

r2 = 4cos(2θ)

Homework Equations



[itex]\frac{dy}{dx}[/itex] = [itex]\frac{r'(θ)sinθ + r(θ)cosθ}{r'(θ)cosθ - r(θ)sinθ}[/itex]

Horizontal Tangent: [itex]\frac{dy}{dθ}[/itex] = 0; [itex]\frac{dx}{dθ}[/itex] ≠ 0
Vertical Tangent: [itex]\frac{dx}{dθ}[/itex] = 0; [itex]\frac{dy}{dθ}[/itex] ≠ 0

The Attempt at a Solution


There is no "clean" way of solving for r (because of the +/- sqrt) so that I could find r'(θ) to use in the formula. So, I figured I would use implicit differentiation:

r2 = 4cos(2θ)

2r[itex]\frac{dr}{dθ}[/itex] = -8sin(2θ)

[itex]\frac{dr}{dθ}[/itex] = [itex]\frac{-4sin(2θ)}{r}[/itex]

But, when I plug it in the formula, I now have an r and a θ when I set the numerator and denominator of [itex]\frac{dy}{dx}[/itex] equal to zero (separately of course). I don't know what to do when I have both variables like that and I'm trying to solve for θ. How should I be approaching this problem? Have I done this problem right so far? If so, what do I do next? If not, any suggestions? Please be clear. I can't find this problem answered clearly anywhere on the internet.
Thank you
 
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  • #2
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  • #3
EnlightenedOne said:
But, when I plug it in the formula, I now have an r and a θ when I set the numerator and denominator of [itex]\frac{dy}{dx}[/itex] equal to zero (separately of course). I don't know what to do when I have both variables like that and I'm trying to solve for θ.
You should have an equation involving r and θ (no derivatives around). Your original equation for the curve also has those two variables. Two equations, two unknowns.
If still stuck, please post all your working.
 

FAQ: Horizontal and Vertical Tangents for Polar Equation

1. What is a horizontal tangent for a polar equation?

A horizontal tangent for a polar equation is a point on the curve where the slope of the tangent line is equal to zero. This means that the curve is neither increasing nor decreasing at that point, and the tangent line is parallel to the x-axis.

2. How do you find the horizontal tangents for a polar equation?

To find the horizontal tangents for a polar equation, you need to take the derivative of the equation with respect to the variable theta. Then, set the derivative equal to zero and solve for theta. The resulting value(s) of theta will give you the coordinates of the horizontal tangent point(s).

3. What is a vertical tangent for a polar equation?

A vertical tangent for a polar equation is a point on the curve where the slope of the tangent line is undefined. This means that the curve is changing direction abruptly at that point, and the tangent line is parallel to the y-axis.

4. How do you find the vertical tangents for a polar equation?

To find the vertical tangents for a polar equation, you need to take the derivative of the equation with respect to the variable theta. Then, set the derivative equal to undefined and solve for theta. The resulting value(s) of theta will give you the coordinates of the vertical tangent point(s).

5. Why are horizontal and vertical tangents important in polar equations?

Horizontal and vertical tangents are important in polar equations because they give us information about the behavior of the curve at specific points. They can help us visualize the shape of the curve, and they are also useful for finding critical points and determining the concavity of the curve.

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