POLAR COORDINATES DEFINITION OF POLAR COORDINATES

Transcription

1 POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand what we are talking about. First, fix an origin (called the pole) and an initial ray from O. Each point P can be located by assigning to it a polar coordinate pair (r, ) where r is the directed distance from O to P and is the directed angle from the initial ray to ray OP. When is positive, then the angle was measured counterclockwise, and when is negative, the angle was measured clockwise. By this fact, a given polar coordinate is not unique. EXAMPLE 1: (see diagram) Sometimes, there are occasions when we would like to allow r to be negative. For example, force on an object in a certain direction. This is why we would say that r is a directed distance.

2 EXAMPLE 2: see diagram EXAMPLE 3: Find all the polar coordinates of the point (-3, - / 4). First of all, let us plot the original point. (Remember we can go around the circle an infinite number of times in either direction, which is why I use +/- and multiples of 2.) Now, if r is positive, what is the that gives the same point? What is the / 4 angle in quadrant II? The answer is 3 / 4. Here are

3 those related points. ELEMENTARY COORDINATE EQUATIONS AND INEQUALITIES Some forms of basic polar equations that occur most often are the equations for a circle centered at the origin and the equation for the line through the origin. FACT: The equation of a circle with radius a centered at the origin is r = a. Why did I say that the radius is a? Remember r is a directed distance, so the circle r = a and r = -a will be the same circle, but start in different places. EXAMPLE 4: To illustrate the above fact I will graph the half circles r = - 2 and r = 2 on the interval [0, ]. r = - 2 is in the blue. r = 2 is in the red. Notice that r = - 2 starts at and goes to 2, whereas r = 2 starts at 0 and goes to. FACT: The equation of a line through the origin making angle o with the initial ray is = o. In my opinion, it is kind of a waste using a polar equation to represent the equation of a line, but it is possible to do. Now, let us graph some polar equations and inequalities. EXAMPLE 5: r 1

4 The graph of this polar inequality will be the shaded region outside the circle of radius 1. EXAMPLE 6: This is the graph of the line that makes the angle 2 / 3 with the positive x-axis, but goes in the opposite direction starting at 2. EXAMPLE 7: 0, r = -1 This is the half circle that starts at and goes to 2. I know that the interval starts at 0, but r is negative. Therefore it goes in the opposite directions. EXAMPLE 8:

5 The graph of this set of inequalities is two wedges cut out of the circle with radii of -1 and 1, and all circles that are between those two values by the lines / 4. CONVERTING FROM POLAR TO CARTESIAN AND VICE VERSA Here are the basic equations that relate polar coordinates to Cartesian coordinates. x = r cos y = r sin x 2 + y 2 = r 2 tan = y/ x Here is a diagram to help us understand where these equations came from. CONVERTING FROM POLAR COORDINATES TO CARTESIAN COORDINATES EXAMPLE 9: Convert (0, / 2) to Cartesian coordinates. x = 0 cos ( / 2) = 0 y = 0 sin ( / 2) = 0 So (0, / 2) is equivalent to (0, 0) in Cartesian coordinates. EXAMPLE 10: Convert the following polar coordinate to its equivalent Cartesian coordinate.

6 So the equivalent Cartesian coordinates for the given polar coordinate is (-1, -1). CONVERTING FROM A CARTESIAN EQUATION TO A POLAR EQUATION EXAMPLE 11: Convert y = 10 into a polar equation. This is a graph of a horizontal line with y-intercept at (0, 10). EXAMPLE 12: Convert x 2 - y 2 = 4 into a polar equation. This is an equation of a hyperbola, and here is its graph.

7 EXAMPLE 13: Convert y 2 = 4x into a polar equation. This is an equation of a parabola, and here is its graph. EXAMPLE 14: Convert (x + 2) 2 + (y - 4) 2 = 16 into a polar equation. First of all, I am going to multiply out the original equation. (x + 2) 2 + (y - 4) 2 = 16 x 2 + 4x y 2-8y + 16 = 16 x 2 + y 2 + 4x - 8y = -4 Now convert this equation into its corresponding polar form. r 2 + 4r cos - 8r sin = -4 This is an equation of a circle with center at (-2, 4) and radius 4. CONVERTING A POLAR EQUATION TO A CARTESIAN EQUATION EXAMPLE 15: Convert r sin = 4 into its equivalent Cartesian equation.

8 r sin = 4 y = 4 This is an equation of a horizontal line through the point (0, 4). EXAMPLE 16: Convert r sin = r cos + 4 into its equivalent Cartesian equation. r sin = r cos + 4 y = x + 4 This is an equation of a line with slope of 1 and y-intercept (0, 4). EXAMPLE 17: Convert r = csc e r cos into its equivalent Cartesian equation. EXAMPLE 18: Convert r = 4tan sec into its equivalent Cartesian equation. In this set of supplemental notes, I defined what makes up a polar coordinate and that a polar coordinate for a point is not unique. Then I talked about the polar equations for circles centered at the origin and lines going through the origin. Finally, I discussed how we could convert from a Cartesian equation to a polar equation by using some formulas. Work through these examples taking note how each conversion was done. Polar coordinates are the first type of coordinates that we will learn in this course and in calculus III. Polar coordinates allow us to graph certain types of curves easily and simplify integrals. In the next three sets of supplemental notes, we will investigate applications of polar coordinates, so make sure that you understand what is happening in this set.