Tangents to a Polar Curve

In summary, the Cartesian equations of the tangent lines at the pole of r=sin2theta are x=0 and y=0, which correspond to horizontal and vertical tangents, respectively. These results may seem confusing without a sketch of the polar graph, but they are correct based on the equations for slope of a polar curve.
  • #1
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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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  • #2
It is a pity that no sketch of the polar graph was posted. The result looks right to me.
 
  • #5
Hm, yes weird, the solutions x=0 and y=0 come from the equation for slope he writes.
 

1. What is a polar curve?

A polar curve is a type of graph that represents a relationship between two variables, usually r and θ, in polar coordinates. It is created by plotting points based on the values of the two variables.

2. How do you find the tangent to a polar curve?

To find the tangent to a polar curve at a certain point, you can use the formula dy/dx = (dy/dθ)/(dx/dθ). This involves finding the first derivatives of the polar equation with respect to θ and then solving for the slope of the tangent at the desired point.

3. Can a polar curve have more than one tangent at a point?

Yes, a polar curve can have multiple tangents at a single point. This usually occurs when the curve has a sharp point or cusp, where the slope of the curve changes abruptly.

4. How do you graph the tangent to a polar curve?

To graph the tangent to a polar curve, you can first plot the point of interest on the curve. Then, using the slope found in the previous step, you can draw a line from that point with the same slope as the tangent. This line represents the tangent to the curve at that point.

5. What is the significance of tangents to a polar curve?

Tangents to a polar curve can help us understand the behavior of the curve at a certain point. They can also be used to find the direction of motion or velocity of a particle moving along the curve. Additionally, tangents can be used to find the slope of the curve and the rate of change of the curve at a given point.

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