Section 10.4 – Polar Coordinates and Polar Graphs

Introduction to Polar CurvesParametric equations allowed us a new way to define relations: with two equations. Parametric

curves opened up a new world of curves:

2cos 2sin 4

x ty t

Polar coordinates will introduce a new coordinate system.

Introduction to Polar CurvesYou have only been graphing with standard Cartesian

coordinates, which are named for the French philosopher-mathematician, Rene Descartes.

Example: Plot

𝑥

𝑦

Polar CoordinatesIn polar coordinates we identify the origin as the pole and the

positive -axis as the polar axis. We can then identify each point in the plane by polar coordinates , where gives the distance from

to and gives the angle from the initial ray to the ray . By convention, angles measured in the counterclockwise direction

are positive.

NOTE: The origin has no well-

defined coordinate. For

our purposes the coordinates will

be for any .

Since it easier to plot a point by starting

with the angle, polar equations are like

inverses. independent variable. dependent variable.

Example 1Example: Plot the polar coordinates .

To plot a point using

polar coordinates , we often use a polar grid:

First find the angle on the

polar grid.

Now plot the point units in

the direction of the angle.

𝟐𝝅𝟑𝟑

Example 2Example: Plot the polar coordinates .

To plot a point using

polar coordinates , we often use a polar grid:

First find the angle on the

polar grid.

Now plot the point units in

the direction of the angle. If is negative, the

point is plotted units in the opposite direction.

𝟓𝝅𝟑

2

Example 3Graph the polar curve . Indicate the direction in which it is traced.

r03cos 0=3

3cos 𝜋6 =2 .598

3cos 𝜋4 =2.121

3cos 𝜋3 =1.5

3cos 𝜋2 =0

3cos 2𝜋3 =−1.5

3cos 3𝜋4 =−2.1

3cos 5𝜋6 =−2.6

3cos 𝜋=−3

Notice Polar equations are like inverses. independent variable.

dependent variable.

The Relationships Between Polar and Cartesian Coordinates

Find the relationships between .(𝑟 , 𝜃)

𝑥

𝑦𝑟

𝑥

𝑦

Right triangles are always a convenient shape to draw.

Using Pythagorean Theorem…

𝑥2+𝑦2=𝑟2

The Relationships Between Polar and Cartesian Coordinates

Find the relationships between .(𝑟 , 𝜃)

𝑥

𝑦𝑟

𝑥

𝑦

What about the angle ?

𝜃

You can use a reference angle to find a relationship but that would require an

extra step.

Instead, compare the coordinates to the unit circle coordinates.

(cos𝜃 , sin 𝜃)

1 The red and blue triangles are similar with a scale factor of . Thus…

cos𝜃sin𝜃

The Relationships Between Polar and Cartesian Coordinates

Find the relationships between .(𝑟 , 𝜃)

𝑥

𝑦𝑟

𝑥=𝑟 cos𝜃

𝑦=𝑟sin𝜃

What about a relationship with ?

𝜃

To find the angle measure , it is possible to use the tangent function to

find the reference angle.Instead investigate the tangent

function and :

tan𝜃=¿sin 𝜃cos𝜃=¿𝑟 sin𝜃𝑟 cos𝜃

Therefore:

tan𝜃=𝑦𝑥

(Remember tangent is also the slope of the radius.)

Conversion Between Polar and Cartesian Coordinates

When converting between coordinate systems the following relationships are helpful to remember:

𝑥𝑦

NOTE: Because of conterminal angles and negative values of r, there are infinite ways to represent a Cartesian Coordinate in Polar Coordinates.

Example 1Complete the following:

a) Convert into polar coordinates.

b) Express your answer in (a) as many ways as you can.

31tan

1 31tan

3

22 21 3r 2 4r

2r

32,

3 32, 2, 2 n

4 43 32, 2, 2 n

Example 2Find rectangular coordinates for .

5616cosx

8 3x

5616siny

8y

8 3,8

NOTE: In Cartesian coordinates, every point in the plane has exactly one ordered pair that

describes it.

Example 3Use the polar-rectangular conversion formulas to show that the polar graph of is a circle.

4sinr r2 4 sinr r

2 2 4x y y 2 2 4 0x y y

2 2 4 4 4x y y

22 2 4x y

A circle centered at (0,2) with a radius of 2

units.

Conversion Between Polar Equations and Parametric Equations

The polar graph of is the curve defined parametrically by:

Example: Write a set of parametric equations for the polar curve

cosx r t sin 6 cost tsiny r t sin 6 sint t

Since we can easily convert a polar equation into parametric equations, the calculus for a polar equation can be performed with the parametrically

defined functions.

The slope of tangent lines is dy/dx not

dr/dΘ.

ExampleUse polar equation to answer the following questions:(a) Find the Cartesian equation of the tangent line

at .

cosx r t 2sin 3 cost t

//

dy dy dtdx dx dt 2sin3 sin

2sin3 cos

ddtddt

t t

t t6cos3 sin 2sin3 cos6cos3 cos 2sin3 sin

t t t tt t t t

6

3dydx t

1 3 3y x

Parametric Equations:

siny r t 2sin 3 sint tFind dy/dx not dr/dΘ:

Find the slope of the tangent line (Remember ):

Find the point:

6 62sin 3 cosx 6 62sin3 siny

31

Find the equation:

Example (Continued)Use polar equation to answer the following questions:(b) Find the length of the arc from to .

cosx r t 2sin 3 cost t

2sin 3 sindy ddt dt t t

6cos3 sin 2sin 3 cost t t t

6 22

0

dydxdt dtd dt

2.227

Parametric Equations:

siny r t 2sin 3 sint tFind dy/dt and dx/dt:

Use the Arc Length Formula:

6 2 2

06cos3 cos 2sin 3 sin 6cos3 sin 2sin3 cost t t t t t t t dt

2sin 3 cosdx ddt dt t t

6cos3 cos 2sin 3 sint t t t

Example (Continued)Use polar equation to answer the following questions:(c) Is the curve concave up or down at .

cosx r t 2sin 3 cost t

//

dy dy dtdx dx dt 2sin3 sin

2sin3 cos

ddtddt

t t

t t 6cos3 sin 2sin3 cos6cos3 cos 2sin3 sin

t t t tt t t t

Parametric Equations:siny r t 2sin 3 sint t

Find dy/dx:

2

2

dd y dtdx

dy dxdx dt

2

2

2 4cos 3 5

3cos3 cos sin3 sin

t

t t t t

Find d2y/dx2:

2

2

6

40d ydx t

Find value of the second derivative (Remember ):

Since the second derivative is

positive, the graph is concave up.

Alternate Formula for the Slope of a Tangent Line of a Polar Curve

If is a differentiable function of , then the slope of the tangent line to the graph of at the point is:

/ ( )cos( ) '( )sin/ ( )sin '( )cos

dy dy d f fdx dx d f f

If you do not want to easily convert a polar equation into parametric equations, you can always memorize

another formula...

Alternate Arc Length Formula for Polar Curves

The arc length for a polar curve between and is given by

22 drdL r d

If you do not want to easily convert a polar equation into parametric equations, you can always memorize

another formula...