10.6 Polar Coordinates10.7 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

222 tan

sin cos

yxrx

y

ryrx

Convert to rectangular coordinates

6

7,2

r = 2θ = 7π/6

6

7sin2

6

7cos2

y

x

Convert to polar coordinates (-4, 1)

4

1tan

x

y

144

1tan 1

90

7 17

17

)1()4(2

222

222

r

r

r

yxr

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 x2 + y2 = r2

• 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/2Polaraxis

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