Demana, Waits, Foley, Kennedy 6.4 Polar Coordinates

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.

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.

Example: Plotting Points in the Polar Coordinate System

Solution

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.

Finding Polar coordinates  

Finding all Polar Coordinates of a Point

Coordinate Conversion Equations

Example: Converting from Polar to Rectangular Coordinates

Solution

Example: Converting from Rectangular to Polar Coordinates

Solution

Example A: Rectangular to Polar  

Example A: Rectangular to Polar  

Example: Converting from Polar Form to Rectangular Form

Solution  

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

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

r = 2sinθ r2 = 2rsinθ (mult each side by r to get r2 rsinθ)   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

Solve and graph: 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

Example: Converting from Polar Form to Rectangular Form

Solution

Solution Continued