Polar Coordinate. December 25, 2016

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1 Polar Coordinate December 25, Introduction Throughout this course, we have been familiar with reading and plotting graph on rectangular coordinates or Cartesian coordinates, with x and y axes. But sometimes it is easier to write the function and plotting the graph on Polar Coordinates of r and θ. Before we continue on this topic, it is necessary to learn about parameterisation first, since Polar Coordinates is just a special type of parameterisation. 2 Parameterisation Parametrics is a technique to plot a graph by specifying the x, y and z values separately. We do so using a parameter, a variable that we chosed to link the equations together. For example, we frequently use the parameter t. For example, if we take x = cos t and y = sin t. Notice that, although we have two separate equations, the x and y are linked by the value of t. In this example, we take a value for the parameter t and plug that value into both equations, to get a corresponding point (x, y). When t = 0, we get (1, 0). So, the point (1, 0) corresponds to the value t = 0. One of the advantages of using parametric equations is that we can describe many more graphs than we could when we had only x and y. Another piece of information we get from parametric equation is direction. The use of the variable t as a parameter is not random. Often, we assign a meaning to the parameter and sometimes that meaning is time. When you graph a set of parametric equations, the graph is swept out in a certain direction. This is an inherent feature of the parametric equations. We will often start at t = 0 and increase t, giving the 1

2 idea that time is passing. By adjusting the parametric equations, we can reverse the direction that the graph is swept. 2.1 Graphing/Sketching Parametric Equations To plot graph, make a table of the parameter t (usually consists of negative, 0, and positive value), x value and y value. It would be easier if you know how to input this into calculator efficiently. Example 1 Sketch the graph for the parametric equations 1. x = 2t + 1, y = t 1 2. x = 2t 2, y = 4t 3. x = 3 + 2t 2, y = 4t 4. x = 3 cos θ, y = 3 sin θ 2.2 Eliminating the Parameter Sometimes we are given the set of parametric equations and we are asked to write the equation without the parameter by eliminating the parameter. This not always possible but with some equations there are ways to do it. The easiest technique to try is to solve one of the equations for the parameter and then substitute the result in the other equation. Here is an example. x = 2t and y = t 2. Solve the first equation for t giving t = x/2 and substitute into the second equation. y = (x/2) 2 = x 2 /4. A second technique involves the use of trig identities. For example, given the parametric equations x = cos t and y = sin t, we know that cos 2 (x) + sin 2 (x) = 1. So we can write x 2 + y 2 = 1 which eliminates the parameter. A third way is by inspection. Sometimes it is obvious what a substitution might be. For example, if we have x = e t and y = e 3t + 1 then we can rewrite y = (e t ) Then we can replace e t with x in the last equation (because our first parametric equation was x = e t ) to get y = x Of course, the first technique would have worked too by solving x = e t for t and then substituting and simplifying but by standing back and looking at the equations more carefully, the solution was much easier. Example 2 Obtain the Cartesian equation by eliminating the parameter from the following paramteric equations. 2

3 1. x = 2t, y = 4t x = 4t, y = 4 t 3. x = 2 m, y = m x = 2 5 cos θ, y = 1 3 sin θ 5. x = 3t 2, y = 6t 3 Polar Coordinates System From basic trigonometri, you know that a point in the plane can be described as (x, y) or as (r cos θ, r sin θ). Comparing these two forms gives you the equations x = r cos θ, y = r sin θ. These equations are used to convert between polar coordinates and rectangular coordinates. Remember from trig that angles can be described in an infinite number of Figure 1: Polar Coordinate System in Cartesian ways, since θ = θ + 2π and θ = θ 2π. It is always best to use the smallest possible angle in the interval ( π, π] or [0, 2π] or whatever is required by the context. Note that these angles are measured in degrees. However, in calculus we almost always specify angles in radians. There are no restriction of the application, you should carefully choose when you input the value in your calculator. Most of the time we use it in degrees since it is more easier to imagine and later plot it in the graph. One of the biggest differences you will find between trigonometri and polar coordinates is that in trig, r in the above equation is usually 1. Trigonometri focuses on the unit circle (when r = 1). However, in polar coordinates we generalize the equations so that r is usually not 1. 3

4 Figure 2: Polar Coordinate plot Figure 3: Trigonometric Quadrants The positive x-axis is called the polar axis, labeled L or sometimes x in Figure 2 and the point O is called the pole. All angles are measured from the polar axis with positive angles in a counter-clockwise direction. In conclusion, polar coordinates are just parametric equations where the parameter is the angle θ and r is a function of θ. Example 3 Plot the following points 1. (3, 225 ) 2. (2, 60 ) 3. (3, 150 ) 4. (4, π 2 ) 5. (1, 120 ) 6. (2, 7π 6 ) 7. (2, π) 4

5 4 Converting Polar Coordinates Cartesian Coordinates In order to convert polar coordinates to Cartesian coordinates, we just need the formula x = r cos θ, y = r sin θ, θ = tan -1 y x, x2 + y 2 = r 2. Example 4 Express the following in polar equations 1. y = x 2 2. x 2 + y 2 = 5 3. x + y = 1 4. xy = a 2 5. x = 4 6. y = 2 7. x 2 + y 2 = 2x 8. x 2 = y Example 5 Express the points in Example 3 in Cartesian coordinates. 5 Graph Sketching of Polar Equation The method to sketch the graph of polar equation is similar to that of Cartesian equations. We start with θ usually from 0 to 2π, or in degrees, with distance r and marked with point in the form (r, θ). Example 6 Sketch the graph for the polar equations given 1. r = 3 2. r = 1 3 π 3. r = 1 cos θ 4. r = 2 + cos θ 5. r = 2 sin θ 5

6 6. r = 3 sin 2θ Knowing if the equation is symmetrical or not will make it easier to draw the graph since repeating calculations can be avoided. Testing the symmetrical is done as below Symmetry Symmetrical Conditions About the x-axis If (r, θ) satisfy the equation, so does (r, θ). About the y-axis If (r, θ) satisfy the equation, so does (r, π θ). About the pole(origin) If (r, θ) satisfy the equation, so does (r, π + θ). Identities that will come in handy : sin( θ) = sin θ, cos( θ) = cos θ, tan( θ) = tan θ. IMPORTANT : It is compulsory for you to test for all axes. It is always the case that it would symmetry on one axis only, OR symmetry on all three axes. Usually, if the polar equation is symmetry on x-axis, it is convenient for you to only look for the value when plotting the graph, then reflect the graph on x-axis. Also, if the polar equation is symmetry on y-axis, it is convenient for you to only look for the value 0 90 AND when plotting the graph, then reflect the graph on y-axis. If the polar equation is symmetry on origin(pole), then you can choose either domain mentioned above, and then rotate 180 of every point that you plot. Coincidentally, if the polar equation is symmetry on the pole, it has the same effect of having to plot the polar coordinate with negative r, for which we plot the positive (r, θ) first then reflect it on origin to get ( r, θ). You might not convince sometimes with your plot after performing the symmetrical test. Therefore, it is okay, for you to tabulate all the value of r from provided you must do the symmetrical test first as required. Example 7 Test the symmetries of the following equations. Hence sketch the graphs. 1. r = 2 + cos θ 2. r = 3 cos 2θ 3. r = 1 sin θ 4. r = 2 3 sin θ 6

7 5. r = a sin θ 6. r 2 = 2a 2 cos 2θ 6 Intersection of Curves in Polar Coordinates If both equation are given in r and θ, just equate the r. But sometimes you need to change it into Cartesian first to find the points of intersection. Example 8 Find the intersection between each of the following pairs of curves. 1. r = 2 cos θ and r = 2 sin θ 2. r = 1 and r 2 = 2 cos 2θ 3. r = a and r = 2a sin θ 4. r = 1 + cos θ and r = 3 cos θ IMPORTANT : It should be noted that, the process of finding the intersection by equating the two polar functions, will not detect certain intersection points due to the polar and angle properties. We usually missed the intersection points that occured on origin or on axes. The best way to rectify this problem is by plotting both of the polar coordinate function in a graph. 7 Conclusion For this topic you are expected to know how to 1. Convert Cartesian coordinates to polar coordinates and vice versa 2. Sketch the graph for polar equations and use symmetrical test 3. Find the intersection points of curves 7