Polar Coordinates

We Live on a Sphere

Polar Coordinates

Up till now, we have graphed on the Cartesian plane using rectangular coordinates

In the rectangular coordinate system a point is plotted as (x, y).

Polar Coordinate

In a polar coordinate system, we select a pint, called to pole, and then a ray with vertex at the pole, called the polar axis.

We still use an ordered pair to graph. The new ordered pair is (r, θ). If r > 0, then r is the distance of the

point from the pole (like the origin)

Polar Coordinates

θ is the angle (in degrees or radians) formed by the polar axis and a ray from the pole.

We call the ordered pair (r, θ) the polar coordinates of the point.

Polar Coordinates

Since angles have several different ways to name them, there are an infinite number of polar coordinates for each point. (Unlike rectangular coordinates which have only one name for point on the Cartesian plane.)

Polar Coordinates

Find four names for the point

We are given a positive radius and a positive angle. We want to find a positive angle and a negative r, a pos r and neg angle, and a neg angle and neg r.

2,4

Steps for finding other polar coordinates

1. Subtract 360o (or 2) to get a negative angle

2. Add 180o (or ) to change the r to negative (half-way around the circle to be on the other side of the polar graph)

3. Add or subtract 360o (or ) to find the other angle

Example

Graph Examples

Conversion from Polar Coordinates to Rectangular Coordinates

If P is a point with polar coordinates (r, ), the rectangular coordinates

(x, y) of P are given by x = r cos θ y = r sin θ

Remember

cos sinx y

r r

Examples

Find the rectangular coordinates of the points with the following polar coordinates:

6, 6cos 6sin6 6 6

3 16 3 3 6 3 3 3, 32 2

x y

x y

Examples( 3, )

3cos 3sin

3 0

(3,0)

32,4

3 2 3 22cos 2 2sin 2

4 2 4 2

2, 2

Polar to Rectangular Coordinates

You can check your answers using your calculator.

First do 2nd APPS Choose Put in polar coordinates Hit enter

(P Rx

Steps for Converting from Rectangular to Polar Coordinates

1. Always plot the point (x, y) first 2. To find r, r2 = x2 + y2 (Look familiar?) 3. To find , remember that we only

know x and y. Therefore, the trig value that we can use involves only x and y – tangent.

1tan tany y

x x

Converting from Rectangular Coordinates to Polar Coordinates

Find polar coordinates of a point whose rectangular coordinates are

a. (0, 3)

b. (2, -2)

c. (-3, 3)

Transforming an Equation from Polar to Rectangular Form

Transform the equation r = cos .

We do not have a formula for just cos but we do have one for r cos .

Multiply both sides by r to get r cos

That gives us r2 = r cos

Transforming an Equation from Polar to Rectangular Form

r2 = x2 + y2 and r cos = x

So, x2 + y2 = x

This is the equation of a circle

Find the answer by completing the square

Transforming an Equation from Polar to Rectangular Form

(x2 – x ) + y2 = 0 (complete the square)

(x2 – x + ¼) + y2 = ¼

(x - ½)2 + y2 = ¼

This is a circle whose center is (½, 0) and whose radius is ½

Transforming an Equation from Rectangular to Polar Form

2 2

2 2

2

2 2 3

2( ) 3

2 3

3

2

x y

x y

r

r

Transforming from Rectangular to Polar Form

y2 = 2x (r sin )2 = 2 r cos r2 sin2 = 2r cos

Examples

More Examples

Tutorials