Polar Coordinates Objective: To look at a different way to plot points and create a graph

Polar Coordinates A polar coordinate system in a plane consists of a fixed point O, called the pole (or origin), and a ray emanating from the pole, called the polar axis. In such a coordinate system we can associate with each point P in the plane a pair of polar coordinates (r, q), where r is the distance from P to the pole and q is an angle from the polar axis to the ray OP. The number r is called the radial coordinate of P and the number q the angular coordinate (or polar angle).

Polar Coordinates The points (6, 45o), (5, 120o), (3,225o), and (4, 330o) are plotted below. If P is the pole, then r = 0, but there is no clearly defined polar angle. We will agree that an arbitrary angle can be used in this case; that is, (0, q) are polar coordinates of the pole for all choices of q.

Polar Coordinates The polar coordinates of a point are not unique. For example, the polar coordinates (1, 315o), (1, -45o), and (1, 675o) all represent the same point. In general, if a point P has polar coordinates (r, q), then (r, q + n360o) and (r, q - n360o) are also polar coordinates of P for any nonnegative integer n. Thus, every point has infinitely many pairs of polar coordinates.

Polar Coordinates As defined above, the radial coordinate r of a point P is nonnegative, since it represents the distance from P to the pole. However, it will be convenient to allow for negative values of r as well.

Polar Coordinates To motivate an appropriate definition, consider the point P with polar coordinates (3, 225o). We can reach this point by rotating the polar axis through an angle of 225 and moving 3 units along the pole, or we can reach the point P by rotating the polar axis through an angle of 45 and then moving 3 units from the pole along the extension of the terminal side.

Relationship Between Polar and Rectangular Coordinates Frequently, it will be useful to superimpose a rectangular xy-coordinate system on top of a polar coordinate system, making the positive x-axis coincide with the polar axis. If this is done, the every point P will have both rectangular coordinates (x, y) and polar coordinates (r, q).

Relationship Between Polar and Rectangular Coordinates Looking at the figure, we can see that leading to the relationship This is how you would change polar coordinates to rectangular coordinates.

Relationship Between Polar and Rectangular Coordinates Looking at the figure, we can also see that This is how you would change rectangular coordinates to polar coordinates.

Example 1 Find the rectangular coordinates of the point P whose polar coordinates are (6, 2p/3).

Example 1 Find the rectangular coordinates of the point P whose polar coordinates are (6, 2p/3). Since r = 6 and q = 2p/3, we have

Example 2 Find the polar coordinates of the point P whose rectangular coordinates are . Since x = -2 and y = , we have Since we are in the second quadrant,

Example 2 Find the polar coordinates of the point P whose rectangular coordinates are . Since x = -2 and y = , we have Since we are in the second quadrant,

Graphs in Polar Coordinates We will now consider the problem of graphing equations in r and q, where q is assumed to be measured in radians. In a rectangular coordinate system the graph of an equation in x and y consists of all points whose coordinates (x, y) satisfy the equation.

Graphs in Polar Coordinates However, in a polar coordinate system, points have infinitely many different pairs of polar coordinates, so that a given point may have some polar coordinates that satisfy an equation and others that do not. Given an equation in r and q, we define its graph in polar coordinates to consist of all points with at least one pair of coordinates (r, q ) that satisfy the equation.

Example 3 Sketch the graphs of (a) r = 1 (b) q = p/4

Example 3 Sketch the graphs of (a) r = 1 (b) q = p/4 (a) For all values of q, the point (1, q) is 1 unit away from the pole. Since q is arbitrary, the graph is the circle of radius 1 centered at the pole.

Example 3 Sketch the graphs of (a) r = 1 (b) q = p/4 (b) For all values of r, the point (r, p/4) lies on a line that makes an angle of p/4 with the polar axis. Positive values of r correspond to points on the line in the first quadrant and negative values of r to points on the line in the third quadrant.

Graphs Equations r = f(q) that express r as a function of q are especially important. One way to graph such an equation is to choose some typical values of q, calculate the corresponding values of r, and then plot the resulting pairs (r, q) in a polar coordinate system.

Example 4 Sketch the graph of r = q (q > 0) in polar coordinates by plotting points.

Example 4 Sketch the graph of r = q (q > 0) in polar coordinates by plotting points. Observe that as q increases, so does r; thus, the graph is a curve that spirals out from the pole as q increases. A reasonably accurate sketch of the spiral can be obtained by plotting points that correspond to values of q that are integer multiples of p/2, keeping in mind that the value of r is always equal to the value of q.

Example 5 Sketch the graph of the equation r = sinq in polar coordinates by plotting points.

Example 5 Sketch the graph of the equation r = sinq in polar coordinates by plotting points. The table below shows the coordinates of points on the graph at increments of p/6. Note that 13 points are listed but we plotted only 7. Many points represent the same place on the graph.

Example 5 Sketch the graph of the equation r = sinq in polar coordinates by plotting points. This is what the graph looks like in a rectangular qr- coordinate system.

Example 5 Observe that the points appear to lie on a circle. We can confirm that this is so by expressing the polar equation r = sinq in terms of x and y. To do this, we multiply the equation through by r to obtain

Graphs Just because an equation r = f(q) involves the variables r and q does not mean that it has to be graphed in a polar coordinate system. When useful, this equation can also be graphed in a rectangular coordinate system. For example, the graphs below are both the graph of r = sinq.

Example 6 Sketch the graph of r = cos2q in polar coordinates.

Example 6 Sketch the graph of r = cos2q in polar coordinates. We will use the graph of r = cos2q in rectangular coordinates to visualize the graph in polar coord.

Families of Rose Curves Equations of the following form are called rose curves. Notice when n is odd it is the number of rose petals. When n is even, there are 2n rose petals.

Symmetry Sometimes symmetry can help with graphing equations in polar coordinates. This leads to the following theorem.

Example 8 Sketch the graph of r = a(1 – cosq) in polar coordinates assuming a to be a positive constant.

Example 8 Sketch the graph of r = a(1 – cosq) in polar coordinates assuming a to be a positive constant. Observe that replacing q with –q does not alter the equation, so we know that the graph will be symmetric to the polar axis. Thus, if we graph the upper half, we have the lower half.

Example 8 Sketch the graph of r = a(1 – cosq) in polar coordinates assuming a to be a positive constant. We will now plot points. This graph is called a cardioid (from the Greek word meaning heart).

Families of Curves Equations with any of the four forms below represent polar curves called limacons (from the Latin word “limax” for a snail-like creature that is commonly called a slug). There are four possible shapes for a limacon that are determined by the ratio a/b. If a = b that is a cardioid.

Graphing Look at the following relationships that will always hold. This will make it easy to graph these figures. Notice how the cosine graph is along the x-axis and the sine is along the y-axis. r = 1 + 2cosq r = 1 – sinq r = 2 – cosq

Families of Circles There are three types of circles. Again, memorize this to make graphing easy.

Families of Spirals A spiral is a curve that coils around a central point. Spirals generally have “left-handed” and “right-handed” versions that coil in opposite directions. Below are some common types of spirals.

Lemniscates The graph of a lemniscate (from the Greek word lemniscos for a looped ribbon resembling the number 8) is pictured below. If it was in terms of sine, it would along the y-axis.

Homework Pages 728-729 1-11 odd 17,19 21-42 multiples of 3