POLAR EQUATIONS

Section 10-4

Polar Coordinates

Given:r: Directed distance from the Polar axis (pole) to point PƟ: Directed angle from the Polar axis to ray OP

),( rP

OInitial ray

Each point P in the plane can be assigned polar coordinates (r, Ɵ), as follows.

r = directed distance from O to PƟ = directed angle, counterclockwise from polar axis to segment OP

Polar Coordinates

Polar Graphs

1) Graph the following polar coordinates:

In general, the point (r, Ɵ) can be written as

(r, Ɵ) = (r, Ɵ + 2nπ)or

(r, Ɵ) = (–r, Ɵ + (2n + 1)π)

where n is any integer. Moreover, the pole is represented by (0, Ɵ), where Ɵ is any angle.

Polar Coordinates

To establish the relationship between polar and rectangular coordinates, let the polar axis coincide with the positive x-axis and the pole with the origin

Because (x, y) lies on a circle ofradius r, it follows that r2 = x2 + y2.

Coordinate Conversion

rryx

y

x

so

so sin

so cos

222

2) Convert to rectangular coordinates

6

5,2

3) Convert to rectangular coordinates

4) Convert to Polar coordinates

32 ,2

3,-3

5) Convert the polar equation to rectangular form

6) Convert the polar equation to rectangular form

4cos r

sincos2

4

r

7) Convert the rectangular equation to polar form

8) Convert the rectangular equation to polar form

3 yx

2xy

The graph of r = a is a circle of radius a centered at zero

Ɵ = α is a Line through O making angle α with the initial ray

Polar Graphs

• Symmetric about the x-axis: if the point (r, Ɵ) lies on the graph, the point (-r, Ɵ) or (r, π-Ɵ) lies on the graph

• Symmetric about the y-axis: if the point (r, Ɵ) lies on the graph, the point (-r, -Ɵ) or (r, π-Ɵ) lies on the graph

• Symmetric about the origin: if the point (r, Ɵ) lies on the graph, the point (-r, Ɵ) or (r, π+Ɵ) lies on the graph

Symmetry

Symmetry

9) Graph without a graphing calculator with the values of Ɵ from 0 to 2π.This curve is called a cardioid.To plot points use

Polar Graphs

sin45 r

sin and cos ryrx

The following are simpler in polar form than in rectangular form. The polar equation of a circle having a radius of a and centered at the origin is simply

Special Polar Graphs

cos sin barbarar

Special Polar Graphs

Special Polar Graphs

Spiral of Archimedes 2

1r

Using the parametric form of dy/dx we have

Slope and Tangent Lines

sincos

cossin

rddr

rddr

ddxd

dy

dx

dy

sin)(sin

cos)(cos

fry

frx

• Horizontal

• Vertical

Horizontal and Vertical Tangent Lines

0 where0 d

dx

d

dy

0 where0 d

dy

d

dx

Cusp at (0, 0)

If

then

Then the lineIs tangent to the pole to the graph of

Tangent Lines at the Pole0)(' where0)( ff

fr

6

5 ,

2 ,

6

pole at the tangents

tan

0cos

0sin

sincos

cossin

ddrddr

rddr

rddr

dx

dy

10) Find the equation of the line tangent to the polar curve

4

3 when ,2sin

r

11) Find the vertical and horizontal tangents fo 20 ,cos1 r

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