Thursday, January 30, 2014

Understanding the Basics of Polar Coordinates

The topic of polar coordinates has been confusing to many students I have worked with in my nearly 14 years of tutoring. But understanding the basics of polar coordinates can make what seems to be a very foreign concept a lot easier to understand and work with. In fact, in some situations, using polar coordinates is easier than using rectangular coordinates. What are polar coordinates? 
A scenario that can be represented in a grid-like fashion such as a layout of a city with parallel roads running north to south and parallel roads running east to west can be represented using a rectangular or Cartesian coordinate system. But how do we explore the symmetry of some of the most beautiful things in nature, things with flowing curves such as the butterfly? Trigonometric functions and a system for plotting points called the polar coordinate system is used.

The foundation of the polar coordinate system is a ray that extends to the right. The ray is called the polar axis. Just like a point on the rectangular coordinate system is represented by the ordered pair (x,y), a points P in the polar coordinate system is represented by an ordered pair (r, Θ). The directed distance from the pole, which is the endpoint of the ray, to P is r. The angle from the polar axis to the line segment from the pole to P is Θ.

For example, a clock is a good representation of a polar coordinate system. Suppose the hour hand is pointing directly to 3 on the clock. That is directly horizontal and represents the polar axis. Now, suppose the minute hand is at the 1, representing the time 3:05. Suppose the length of the minute hand is 5 inches. Therefore, r = 5. The angle from the hour hand to the minute hand is Θ. In this case since there are 12 numbers around the clock and a circle is 360 degrees, each number is 30 degrees (360 divided by 12 equals 30). There are two numbers between 3 and 1, so the angle Θ in this example is 60. The polar coordinate of this situation is (5, 60). If using radians, Θ = Pi/3.

Suppose the polar coordinates of a point P are (2, 45) or (2, Pi/4). Since the angle is positive, first draw the 45 degree angle counterclockwise from the polar axis. Then count 2 units along the terminal side of the angle to reach the point P.

The sign of r is important in plotting the polar coordinate P. If r > 0, the point lies on the terminal side of Θ. If r < 0, the point lies opposite the terminal side.

Let's take another example, P = (-3, 270) or (-3, 3Pi/2). First move counterclockwise 270 degrees, which would be a ray pointing straight down. If this was a rectangular coordinate system, the coordinate would down the y-axis. If r equaled 3, the ray would point down three units. But since r is -3, the ray points in the opposite direction 3 units in length. The ray would be pointing straight up three units along the 90 degree or Pi/2 axis. If this was a rectangular coordinate system, the point would be up the y-axis.

An interesting thing about polar coordinates is that each point can be represented infinitely many ways. You can add 2Pi or 360 degrees to Θ and it does not change it's location since it just moves the ray one complete revolution. Also you can change the sign of r and add a half a revolution, 180 degrees or Pi, and that will also not change the location. With rectangular coordinates, there is only one way each point can be represented.
There are basic relations between polar coordinates and rectangular coordinates. They are as follows:

  1. x = rcosΘ, y = rsinΘ
  2. x^2 + y^2 = r^2
  3. tanΘ = y/x
We use the relations to convert from polar coordinates to rectangular coordinates and vice versa. Suppose we have the polar coordinate (2, 3Pi/2) or (2, 270). We can convert to rectangular coordinates by substituting the given values for r and Θ into x = rcosΘ and y = rsinΘ. When doing so, we get x=0 and y = -2. Therefore the rectangular coordinates are (0, -2). 
When converting from rectangular to polar coordinates, first plot the point (x,y). Find r by using the formula x^2 + y^2 = r^2. Find Θ by using tanΘ = y/x with the terminal side passing through (x, y). For example, suppose the rectangular coordinates are (-2,0). After plotting the point we calculate r, -2^2 + 0^2 = r^2. Therefore 4 = r^2, so r = 2. Now we find the angle by taking tanΘ = 0/-2, therefore Θ = 180 or Pi. The polar coordinate P is (2, 180) or (2, Pi).

There are many other complex problems and applications with polar coordinates, but understanding the basics are crucial before moving forward on those types of problems. This guide should help you better understand the basics of polar coordinates.

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