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6.4

Explore the Polar Coordinate System, Equation Conversion, Distance Calculation, and more. Learn how to convert between polar and rectangular coordinates, simplifying complex equations and enhancing calculus skills.

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6.4

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  1. 6.4 Demana, Waits, Foley, Kennedy Polar Coordinates

  2. What you’ll learn about • Polar Coordinate System • Coordinate Conversion • Equation Conversion • Finding Distance Using Polar Coordinates … and why Use of polar coordinates sometimes simplifies complicated rectangular equations and they are useful in calculus.

  3. The Polar Coordinate System A polar coordinate system is a plane with a point O, the pole, and a ray from O, the polar axis, as shown. Each point P in the plane is assigned polar coordinates as follows: r is the directed distance from O to P, and  is the directed angle whose initial side is on the polar axis and whose terminal side is on the line OP.

  4. Example: Plotting Points in the Polar Coordinate System

  5. Solution

  6. Finding Polar Coordinates • Starting at the pole, give four different polar coordinates that would require you to turn 600 (or π/3 radians) and travel a distance of 10 units.

  7. Finding Polar coordinates

  8. Finding all Polar Coordinates of a Point

  9. Coordinate Conversion Equations

  10. Example: Converting from Polar to Rectangular Coordinates

  11. Solution

  12. Example: Converting from Rectangular to Polar Coordinates

  13. Solution

  14. Example A: Rectangular to Polar

  15. Example A: Rectangular to Polar

  16. Example: Converting from Polar Form to Rectangular Form

  17. Solution

  18. Polar to rectangular • Express r = 5 sec in rectangular coordinates and graph the equation

  19. Express r = 5 sec in rectangular coordinates and graph the equation Solution r = 5 secθ r = 5(1/cosθ) r cos θ = 5 x = 5

  20. r = 2sinθ • r2 = 2rsinθ(mult each side by r to get r2 rsinθ) • x2 + y2 = 2y • x2 + y2 - 2y = 0 (complete the square) • x2 + (y - 1)2 = 1 • This is a circle centered at (0, 1) with a radius of 1

  21. Solve and graph: r = 2 + 2cosθ • r = 2 + 2cosθ • r2 = 2r + 2rcosθ (Substitute) • x2 + y2 = 2r + 2x • x2 + y2 -2x = 2r (square each side to get r2) • (x2 + y2 -2x)2 = 4r2 • (x2 + y2 -2x)2 = 4(x2 + y2) (STOP HERE) • x4 - 4x3 + 2x2y2 +4x2 - 4xy2 + y4 = 4x2 + 4y2 • x4 - 4x3 + 2x2y2 - 4xy2 - 4y2+ y4 = 0

  22. Example: Converting from Polar Form to Rectangular Form

  23. Solution

  24. Solution Continued

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