9.6 Polar Coordinates

Objectives Plot points and find multiple representations of points in the polar coordinate system Convert points from rectangular to polar form and vice-versa Convert equations from rectangular to polar form and vice- versa

Rectangular Coordinate Plane point (x, y) in the rectangular plane represent the directed distance from the coordinate axes to the point Each point (x, y) has a unique representation

Polar Coordinate System To form the polar coordinate system in the plane: fix a point O (called the pole or origin) construct from O an initial ray (polar axis) each point P in the plan can be assigned polar coordinates (r, θ) P = (r, θ) r = directed distance from O to P θ= directed angle, counterclockwise from the polar axis to segment OP r=directed distance θ=directed angle Polar Axis

Plotting Points in the Polar Coordinate System (2, π/3) (1, 2π/3) 2. (3, -π/6) 3. (3, 11π/6)

Polar Coordinates – Multiple Representations Unlike rectangular coordinates, polar coordinates are NOT unique the point (r, θ) and (r, θ + 2π) represent the same point the point (r, θ) and (-r, θ + π) are the same In general, (r, θ) = (r, θ ± 2nπ) (-r, θ ± (2n+1)π) where n is any integer The pole is (o, θ), where θ is any angle

Multiple Representations of Points Plot the point (3, -3π/4) Find 3 additional polar coordinates representations of this point using -2π<θ<2π

Coordinate Conversion (x, y) lies on a circle of radius r, it follows that r2 = x2 + y2 coordinate conversion the polar coordinates (r, θ) are related to the rectangular coordinates (x, y) as follows x = r cosθ y = r sinθ tanθ = y/x r2 = x2 + y2

Convert: Polar  Rectangular Convert each point to rectangular coordinates (2, π) (√3, π/6)

Convert: Rectangular  Polar Convert each point to polar coordinates. (-1, 1) (0,2)