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Intersection of Graphs of Polar Coordinates

Explore strategies for finding points of intersection between polar coordinate graphs and determining areas of intersecting regions. Learn how to identify points of intersection by graphing, using substitution, and simultaneous solutions. Practice with exercises on page 455, focusing on r=1 and r=2.cosθ equations.

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Intersection of Graphs of Polar Coordinates

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  1. Intersection of Graphs of Polar Coordinates Lesson 10.9

  2. Why??!! • Lesson 10.10 will be finding area of intersecting regions • Need to know where the graphs intersect • r = 1 • r = 2 cos θ

  3. Strategies • r = 1 • r = 2 cos θ • Use substitution • Let r = 1 in the second equation • Solve for θ • Let @n1 = 0, result is

  4. A Sneaky Problem • Consider r = sin θand r = cos θ • What is simultaneoussolution? • Where sin θ = cos θ that is • Problem … the intersection at the pole does not show up using this strategy • You must inspect the graph

  5. Hints • Graph the curves on your calculator • Observe the number of intersections • Zoom in as needed • Do a simultaneous solution to the two equations • Check results against observed points of intersection • Discard duplicates • Note intersection at the pole that simultaneous solutions may not have given

  6. The others are duplicates Try These • Given r = sin 2θ and r = 2 cos θ • Find all points of intersection • By observation one point is (0, 0) • Use algebra to find the others

  7. Assignment • Lesson 10.9 • Page 455 • Exercises 1 – 11 odd

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