6.4 Polar Equations

$$r=3\ \cos\ \theta$$

$$r\ =\ 3\ \sin\ \theta$$

$$r\ =\ 3\ \cos\ 2\theta$$

$$r\ =\ \sin\ 3\theta$$

$$r=3\ \cos\ \theta$$

$$r\ =\ 3\ \sin\ \theta$$

$$r\ =\ 3\ \cos\ 2\theta$$

$$r\ =\ \sin\ 3\theta$$

$$r\ =\ 3$$

$$r\ =\ \theta$$

$$r\ =\ 7.5$$

$$r\ =\ 3\theta$$

$$r\ =\ 3$$

$$r\ =\ \theta$$

$$r\ =\ 7.5$$

$$r\ =\ 3\theta$$

lemniscate

rose curve

cardioid

sine curve

4

5

72

90

k = 1

k = 2

k = 3

k = 4

$$r\ =\ 0.5\ \sin\ 2\theta$$

$$r\ =\ 0.5\ \cos\ 2\theta$$

$$r\ =\ 3\ \sin\ 2\theta$$

$$r\ =\ \sin\ 4\theta$$

Rose with 4 leaves, Length 4

Rose with 8 leaves, Length 4

Cardioid with length 3

Inner loop limacon with lengths 3 and 4

9

18

8

17

lemniscate

limacon

cardioid

circle centered at the pole

the polar axis

the line $$\theta=\frac{\pi}{2}$$

the pole

r² = 9cos2θ

r² = 9sin2θ

r² = 3cosθ

r² = 3sinθ

lemniscate

rose curve

cardioid

limacon

About the Pole

Over polar axis

Over θ=π/2

No Polar Symmetry

r = 2 + sin θ

r = 2 - sin θ

r = 2 + cos θ

r = 1 + 2sin θ

r = 3 + 2 cosθ

r = 2 + 3 cosθ

r = 2 + 2cosθ

r = 2 - 3cosθ