Polar Coordinates 6.3

6.3 Polar Coordinates Objectives Plot points in the polar coordinate system Find multiple sets of polar coordinate for a given point Convert a point from polar to rectangular coordinates Convert a point from rectangular to polar coordinates Convert an equation from rectangular to polar coordinates Convert an equation from polar to rectangular coordinates

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 .

What is the relationship between cartesian coordinates & polar ones? The radius = r is the hypotenuse of a rt. triangle that has base = x & height=y Thus, If x = horizontal leg & y = vertical leg of a right triangle, then

Text Example Plot the points with the following polar coordinates: a. (2, 135°) 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. (2, 135°) 135º

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

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

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

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

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).

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

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

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

Convert to Rectangular Form:

Convert to Rectangular Form:

Convert to Rectangular Form:

Convert to Rectangular Form: