Results 1 to 4 of 4

Math Help - Points of intersection

  1. #1
    Junior Member
    Joined
    Oct 2007
    Posts
    51

    Points of intersection

    Find the points of intersection (r,theta) of the curves. Points intersect at the pole and one other point.

    r=2(1-cos(theta))
    r=4sin(theta)

    How do you find theta? I know the one of the points is zero but i'm looking for the other point of intersection. I also know what r and theta is if i graph these equations but how do you work it out without a calculator? Details would be appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,111
    Thanks
    2
    One way is to solve for r, rather than \theta.

    It's not necessarily obvious. You must think about it. Sometimes you must try new things.

    From the cosine equation: \theta\;=\;acos\left(\frac{r-2}{2}\right)

    Substitute into the sine equation - and simplify: r\;=\;2\sqrt{4r-r^{2}}

    You should be able to get that far. Only a little algebra remains.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by TKHunny View Post
    One way is to solve for r, rather than \theta.

    It's not necessarily obvious. You must think about it. Sometimes you must try new things.

    From the cosine equation: \theta\;=\;acos\left(\frac{r-2}{2}\right)

    Substitute into the sine equation - and simplify: r\;=\;2\sqrt{4r-r^{2}}

    You should be able to get that far. Only a little algebra remains.
    general Warning: When finding the intersection points of two curves defined in polar coordinates, graphs should ALWAYS be drawn because the solution to the simultaneous equations does not always give ALL intersection points. This is because there is NOT a one-to-one correspondence between points and coordinates in the polar coordinate system.


    For example, the intersection point at the pole of r = \sqrt{3} \cos \theta and r = \sin \theta does NOT show as a solution to the simultaneous equations. This is because the pole is not on the two curves ‘simultaneously’ in the sense of being reached at the same value of \theta: on the curve r = \sin \theta the pole is reached when \theta = 0, whereas on the curve r = \sqrt{3} \cos \theta the pole is reached when \theta = \frac{\pi}{2}. (This can be compared with parametric equations - although the paths followed by two particles may intersect, the particles themselves do not necessarily meet).

    Example: The curves r^2 = 4 a^2 \cos \theta and r = a(1 - \cos \theta ) intersect at FOUR points:

    1. The two points with polar coordinates [r_1, \theta_1] where r_1 = 2a(\sqrt{2} - 1) and \cos \theta_1 = 3 - 2\sqrt{2}, are found from the simultaneous solution.

    2. The two points with Cartesian coordinates (0, 0) and (-2a, 0) are disclosed only by graphing the curves.

    The Cartesian point (0, 0) has polar coordinates [0, \frac{\pi}{2}] on the curve r^2 = 4 a^2 \cos \theta and polar coordinates [0, 0] on the curve r = a(1 - \cos \theta ). Clearly it is not a simultaneous solution in polar coordinates.

    Similarly, the Cartesian point (-2a, 0) has polar coordinates [-2a, 0] on the curve r^2 = 4 a^2 \cos \theta and polar coordinates [0, \pi] on the curve r = a(1 - \cos \theta ).
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    May 2007
    Posts
    237
    It is the intersectin of a cardioid and a circle.
    Use the fact that 1-cos(t)=2sin(t/2)^2 and sin(t)=2sin(t/2)cos(t/2).
    Attached Thumbnails Attached Thumbnails Points of intersection-circle.jpg  
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. points of intersection
    Posted in the Algebra Forum
    Replies: 11
    Last Post: December 23rd 2011, 06:00 PM
  2. Intersection points
    Posted in the Geometry Forum
    Replies: 2
    Last Post: March 25th 2010, 05:56 PM
  3. Points of Intersection
    Posted in the Algebra Forum
    Replies: 1
    Last Post: January 16th 2010, 06:10 PM
  4. points of intersection.
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: September 26th 2008, 04:18 PM
  5. points of intersection
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: August 6th 2007, 10:33 PM

Search Tags


/mathhelpforum @mathhelpforum