Polar Coordinates in Geometry
Polar coordinates are an alternative way of representing points in space. In contrast to Cartesian coordinates, which uses a pair of perpendicular axes (the x-axis and y-axis) to pinpoint locations, polar coordinates express points using a single radial axis (the r-axis) and a rotational axis (the theta-axis).
Polar coordinates can be useful for graphing certain types of functions or for representing points in space that don't lend themselves well to the traditional x- and y-axes. For example, the path of a projectile can be more easily represented using polar coordinates than Cartesian coordinates. In this blog post, we'll take a closer look at how polar coordinates work and how they can be used to graph points and functions.
How Polar Coordinates Work
As we mentioned earlier, there are two components to every polar coordinate: the radial coordinate (r) and the angular coordinate (theta). The radial coordinate corresponds to the distance from the origin (0,0), while the angular coordinate corresponds to the angle between the positive x-axis and the line segment connecting the point to the origin. This angle is measured in degrees or radians; most commonly, radians are used in mathematics and physics applications.
One way to think about it is that the radial coordinate corresponds to how "far out" from the origin a point is, while the angular coordinate corresponds to what "direction" from the origin that point is. Let's take a look at an example to see how this works in practice.
Example: Converting from Cartesian to Polar Coordinates
Suppose we have the point (3,4). To convert this point from Cartesian coordinates to polar coordinates, we need to calculate the radial coordinate and the angular coordinate.
The radial coordinate is simply the distance from the origin to the point; in other words, it's the length of the line segment connecting the origin to the point. We can use the Pythagorean theorem to calculate this:
r = sqrt((3-0)^2 + (4-0)^2)
r = 5
Now, we need to calculate the angular coordinate. Recall that this is the angle between the positive x-axis and the line segment connecting the point to the origin. We can use trigonometry to calculate this:
theta = arctan((4-0)/(3-0))
theta = 53.13 degrees
Therefore, the polar coordinates of our example point are (5, 53.13 degrees).
Graphing Points in Polar Coordinates
Now that we know how to convert between Cartesian and polar coordinates, let's take a look at how to graph points using polar coordinates.
There are two common ways to graph points in polar coordinates. The first is called the "rectangular method," and it's simply a matter of plotting the point with its corresponding r and theta values. So, for our example point (5, 53.13 degrees), we would plot a point at (5,53.13) on a graph.
The second method is called the "polar method." To use this method, we start by plotting the origin (0,0). Then, we use a ruler to draw a line segment from the origin outwards at the angle corresponding to theta. Finally, we measure the length of this line segment and mark it off at the point corresponding to r.
For our example point (5, 53.13 degrees), the polar method would look like this:
We start by plotting the origin (0,0).
Next, we draw a line segment from the origin at an angle of 53.13 degrees.
Finally, we measure the length of this line segment and mark it off at 5 units. This gives us our point (5, 53.13 degrees).
Polar Coordinates and Graphs of Functions
We can also use polar coordinates to graph functions. To do this, we simply plot a series of points that satisfy the function's equation and then connect these points with a smooth curve.
Let's take a look at an example.
Example: Graphing a Function in Polar Coordinates
Suppose we want to graph the following function in polar coordinates:
r = 2 + cos(theta)
To do this, we'll need to calculate a few points that satisfy the equation and then plot these points on a graph. Let's start by finding three points that satisfy the equation.
If we set theta = 0, we get r = 2 + cos(0) = 2 + 1 = 3. So, our first point is (3, 0).
If we set theta = pi/2, we get r = 2 + cos(pi/2) = 2 + 0 = 2. So, our second point is (2, pi/2).
If we set theta = pi, we get r = 2 + cos(pi) = 2 - 1 = 1. So, our third point is (1, pi).
Now that we have our three points, we can plot them on a graph and connect them with a smooth curve. This gives us the following graph:
As you can see, this is a pretty simple graph. However, we can also use polar coordinates to graph more complicated functions.
Polar coordinates are a very useful tool in mathematics and can be used to graph a wide variety of functions. Hopefully this introduction has given you a good understanding of how they work and how to use them.