What is the Equation for Tangents to a Polar Curve at the Pole?

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Homework Help Overview

The discussion revolves around finding the Cartesian equations of tangent lines at the pole of the polar curve defined by r=sin(2θ). The original poster attempts to determine the angles at which the curve intersects the pole and calculate the slopes of the tangents using the polar slope formula.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster calculates the angles where r=0 and applies the polar slope formula, but expresses confusion regarding the resulting horizontal and vertical tangents. Other participants question the absence of a sketch and discuss the nature of the curve, noting it has four petals.

Discussion Status

The discussion is ongoing, with some participants affirming the correctness of the original poster's calculations while others express confusion about the graphical representation of the tangents. There is no explicit consensus on the interpretation of the results.

Contextual Notes

Participants note the lack of a sketch of the polar graph, which may be contributing to the confusion regarding the tangents at the pole. The original poster's calculations lead to results that seem counterintuitive when visualizing the curve.

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Find the Cartesian equations of the tangent lines at the pole of r=sin2theta.* I know to set r=sin2theta equal to zero. This means that theta can equal 0, pi/2, pi, 3pi/2, 2pi. Now I also know to plug each of these theta values into the equation for slope of a polar curve, which is:

r'sintheta + rcostheta/ (r'costheta- rsintheta)

r'= 2cos2theta

*So starting with 0, for example, gives you (2)(0) + (0)(1)/ ((2)(1)- (0)(0). This equals 0/2, which means its a horizontal tangents. I get similar results for the other theta values- either horizontal or vertical tangents. Using x=rcostheta for vertical tangents, @ 0 i get x=0 (and similar results for the other values). I'm so confused though- this doesn't make sense, I looked at the polar graph and I can't see how the tangents at the pole are x=0 and y=0! What am I doing wrong? Any help would be very much appreciated! :-)
 
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It is a pity that no sketch of the polar graph was posted. The result looks right to me.
 
Hm, yes weird, the solutions x=0 and y=0 come from the equation for slope he writes.
 

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