10.6 Polar Coordinates 10.7 Graphs of Polar equations

Polar coordinates An initial ray (polar axis) from a fixed point (the pole or origin); (r, θ) r = directed distance from O to P θ = directed angle, counterclockwise from the polar axis

Graphing polar coordinates r is the radius of the circles that make up the graph θ is the directed angle from the positive x axis

Since polar coordinates are on the unit circle there are multiple representations for one point (r, θ) = (r, θ+/- 2nπ) (r, θ) = (-r, θ+/- (2n+1)π) The pole is represented by (0, θ) where θ is any angle To convert polar to rectangular coordinates

Convert to rectangular coordinates r = 2 θ = 7π/6

Convert to polar coordinates (-4, 1)

Equation Conversion To convert rectangular to polar form use x = rcosθ and y = rsinθ When given polar form r = c (c is a real number) the rectangular equation is a circle of radius c so x 2 + y 2 = r 2 When given θ=c use tan θ = y/x When given r = a trig function, convert the trig function to sin or cos

Graphs of Polar equations Change MODE on calculator to POL Y = is now r= Use the table to plot points where r is the horizontal axis and θ is the vertical axis In the window you can set max/min for θ Use TRACE to find the maximum r-value

Tests for Symmetry The line θ = π/2; replace (r, θ) by (r, π-θ) or (-r, -θ) The polar axis; replace (r, θ) by (r, -θ) or (-r, π-θ) The pole; replace (r, θ) by (r, π+θ) or (-r, θ) θ=π/2 Polar axis Pole

Analyzing the curve Use the chart on page 750 to identify the type of curve Identify the type of symmetry Find the maximum r value Find the zeros of r