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Polar Coordinates

Polar Coordinates. The Sign of r and a Point’s Location in Polar Coordinates.

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Polar Coordinates

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  1. Polar Coordinates

  2. The Sign of r and a Point’s Location in Polar Coordinates. • The point P= (r, ) is located |r| units from the pole. If r > 0, the point lies on the terminal side of . If r < 0 the point lies along the ray opposite the terminal side of . If r= 0. the point lies at the pole, regardless of the value of .

  3. Solution a. To plot the point (r, ) = (2, 135°), begin with the 135° angle. Because 135° is a positive angle, draw = 135° counterclockwise from the polar axis. Now consider r= 2. Because r > 0, plot the point by going out two units on the terminal side of . Figure (a) shows the point. 135º (2, 135°) (2, 135°) Text Example • Plot the points with the following polar coordinates: a. (2, 135°)

  4. If n is any integer, the point (r, ) can be represented as (r, ) = (r, + 2n) or (r, ) = (-r, ++ 2n ) Multiple Representation of Points

  5. Find another representation for in which: • r is positive and • r is negative and • r is positive and

  6. P =(r, )=(x, y) Relations between Polar and Rectangular Coordinates x = r cos y = r sin x2+y2=r2 tan =y/x

  7. Find the rectangular coordinates of the points with the following polar coordinates: a. b.

  8. Converting a Point from Rectangular to Polar Coordinates(r > 0 and 0 < < 2) • Plot the point (x, y). • Find r by computing the distance from the origin to (x, y). • Find  using tan = y/x with  lying in the same quadrant as (x, y).

  9. Example • Find the polar coordinates of a point whose rectangular coordinates are (2, 4) Solution:

  10. Example • Find the polar coordinates of a point whose rectangular coordinates are (-1, √3) Solution:

  11. Example • Convert 2x-y=1 to a polar equation. Solution:

  12. Convert to Rectangular Form:

  13. Convert to Rectangular Form:

  14. Convert to Rectangular Form:

  15. Convert to Rectangular Form:

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