Week 12, Day 1

Today we learned about polar coordinates. As opposed to the traditional x-y coordinate system, we use the polar coordinate system. To form this system in the plane, fix a point O, called the pole or origin, and construct from O an initial ray called the polar axis. Then each point P in the plane can be assigned polar coordinates (r, theta) as follows. (1) r = directed distance from O to P (2) Theta = directed angle, counterclockwise from polar axis to segment OP. 

Example 1: Plotting Points in the Polar Coordinate System

a) The point (r, theta) = (2, pi/3) lies two units from the pole on the terminal side 

of the angle theta = pi/3, as shown in the following image. 

b) The point (r, theta) = (3, -pi/6) lies three units from the pole on the terminal side 

of the angle theta = -pi/6, as shown in the following image. 

c) The point (r, theta) = (3, 11pi/6) coincides with the point (3, -pi/6), as shown in the following image. 

In rectangular coordinates, each point (x,y) has a unique representation. This is not true for polar coordinates. For instance, the coordinate (r, theta) and (r, 2pi + theta) represent the same point. Another way to obtain multiple reoresentations of a point is to use negative values for r. Because r is a directed distance, the coordinates (r, theta) and (-r, theta + pi) represent the same point. In general, the point (r, theta) can be represented as (r, theta) = (r, theta +/- 2npi) or (r, theta) = (-r, theta +/- (2n+1)pi) where n os any interger. Moreover, the pole is represented by (0, theta), where theta is any angle. 

Example 2: Multiple Representation of Points

Plot the point (3, -3pi/4) and find three additional polar representations of this point, using -2pi < theta < 2pi. The point is shown in the following image. 

Three other representations are as follows.

(3, -3pi/4 + 2pi) = (3, 5pi/4): Add 2pi to theta.

(-3, -3pi/4 - pi) = (-3, -7pi/4): Replace r by -r, then subtract pi from theta.

(-3, -3pi/4 + pi) = (-3, pi/4): Replace r by -r, then add pi to theta.

Coordination Conversion

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 of radius r, it follows that r^2 = x^2 + y^2. Moreover, for r > 0, the defintions of the trignometric functions imply that tan theta = y/x, cos theta = x/r, and sin theta = y/r. You can show that the same relationships hold for r > 0. 

The polar coordinates (r, theta) are related to the rectangular coordinates (x,y) as follows:

x = rcos theta and tan theta = y/x

y = rsin theta 

r^2 = x^2 + y^2

Example 3: Polar-Rectangular Conversion

Convert the points (a) (2, pi) and (b) ( (3)^1/2, pi/6) to rectangular coordinates.

(a) For the point (r, theta) = (2, pi), you have

x = rcos theta = 2 cos pi = -2 and y = rsin theta = 2 sin pi = 0.

The rectangular coordinates are (x,y) = (-2,0).

(b) For the point (r, theta) = ( (3)^1/2, pi/6), you have

x = (3)^1/2 cos pi/6 = (3)^1/2 ( (3)^1/2 / 2) = 3/2 and

y = (3)^1/2 sin pi/6 = (3)^1/2 (1/2) = (3)^1/2 / 2.

The rectangular coordinates are (x,y) = ( 3/3, (3)^1/2).

Example 5: Converting Polar Equations to Rectangular Form

Describe the graph of each polar equation and find the corresponding rectangular equation.

(a) r = 2 

The graph of the polar equation r = 2 consists of all points that are two units from the pole. In other words, this graph is a circle centered at the origin with a radius of 2. You can confirm this by converting to rectangular form, using the relationship r^2 = x^2 + y^2. 

r = 2 is the polar equation. 

r ^2 = 2^2 

x^2 + y^2 = 2^2 is the rectangular equation.