9.1 Polar Coordinates Contemporary Calculus 1 9.1 POLAR COORDINATES The rectangular coordinate system is immensely useful, but it is not the only way to assign an address to a point in the plane and sometimes it is not the most useful. In many experimental situations, our location is fixed and we or our instruments, such as radar, take readings in different directions (Fig. 1); this information can be graphed using rectangular coordinates (e.g., with the angle on the horizontal axis and the measurement on the vertical axis). Sometimes, however, it is more useful to plot the information in a way similar to the way in which it was collected, as magnitudes along radial lines (Fig. 2). This system is called the Polar Coordinate System. In this section we introduce polar coordinates and examine some of their uses. We start with graphing points and functions in polar coordinates, consider how to change back and forth between the rectangular and polar coordinate systems, and see how to find the slopes of lines tangent to polar graphs. Our primary reasons for considering polar coordinates, however, are that they appear in applications, and that they provide a "natural" and easy way to represent some kinds of information. Example 1: SOS! You've just received a distress signal from a ship located at A on your radar screen (Fig. 3). Describe its location to your captain so your vessel can speed to the rescue. Solution: You could convert the relative location of the other ship to rectangular coordinates and then tell your captain to go due east for 7.5 miles and north for 13 miles,9.1 Polar Coordinates Contemporary Calculus 2 but that certainly is not the quickest way to reach the other ship. It is better to tell the captain to sail for 15 miles in the direction of 60°. If the distressed ship was at B on the radar screen, your vessel should sail for 10 miles in the direction 150°. (Real radar screens have 0° at the top of the screen, but the convention in mathematics is to put 0° in the direction of the positive x–axis and to measure positive angles counterclockwise from there. And a real sailor speaks of "bearing" and "range" instead of direction and magnitude.) Practice 1: Describe the locations of the ships at C and D in Fig. 3 by giving a distance and a direction to those ships from your current position at the center of the radar screen. Points in Polar Coordinates To construct a polar coordinate system we need a starting point (called the origin or pole) for the magnitude measurements and a starting direction (called the polar axis) for the angle measurements (Fig. 4). A polar coordinate pair for a point P in the plane is an ordered pair (r,θ) where r is the directed distance along a radial line from O to P, and θ is the angle formed by the polar axis and the segment OP (Fig. 4). The angle θ is positive when the angle of the radial line OP is measured counterclockwise from the polar axis, and θ is negative when measured clockwise. Degree or Radian Measure for θ? Either degree or radian measure can be used for the angle in the polar coordinate system, but when we differentiate and integrate trigonometric functions of θ we will want all of the angles to be given in radian measure. From now on, we will primarily use radian measure. You should assume that all angles are given in radian measure unless the units " ° " ("degrees") are shown. Example 2: Plot the points with the given polar coordinates: A(2, 30°), B(3, π/2), C(–2, π/6), and D(–3, 270°). Solution: To find the location of A, we look along the ray that makes an angle of 30° with the polar axis, and then take two steps in that direction (assuming 1 step = 1 unit). The locations of A and B are shown in Fig. 5. To find the location of C, we look along the ray which makes an angle of π/6 with the polar axis, and then we take two steps backwards since r = –2 is negative. Fig. 6 shows the locations of C and D. Notice that the points B and D have different addresses, (3, π/2) and (–3, 270°), but the same location.9.1 Polar Coordinates Contemporary Calculus 3 Practice 2: Plot the points with the given polar coordinates: A(2, π/2), B(2, –120°), C(–2, π/3), D(–2, –135°), and E(2, 135°). Which two points coincide? Each polar coordinate pair (r,θ) gives the location of one point, but each location has lots of different addresses in the polar coordinate system: the polar coordinates of a point are not unique. This nonuniqueness of addresses comes about in two ways. First, the angles θ, θ ± 360°, θ ± 2.360°, . . . all describe the same radial line (Fig. 7), so the polar coordinates (r, θ), (r, θ ± 360°), (r, θ ± 2.360°) , . . . all locate the same point. Secondly, the angle θ ± 180° describes the radial line pointing in exactly the opposite direction from the radial line described by the angle θ (Fig. 8), so the polar coordinates (r, θ) and (–r, θ ± 180°) locate the same point. A polar coordinate pair gives the location of exactly one point, but the location of one point is described by many (an infinite number) different polar coordinate pairs. Note: In the rectangular coordinate system we use (x, y) and y = f(x): first variable independent and second variable dependent. In the polar coordinate system we use (r, θ) and r = f(θ): first variable dependent and second variable independent, a reversal from the rectangular coordinate usage. Practice 3: Table 1 contains measurements to the edge of a plateau taken by a remote sensor which crashed on the plateau. Fig. 9 shows the data plotted in rectangular coordinates. Plot the data in polar coordinates and determine the shape of the top of the plateau. distance (feet)1020300°90° 180° 270°360°angle (degrees)Rectangular Coordinate Graph of Data4040°130°210°340°Fig. 9 angle distance0° 28 feet20° 30 40° 36 60° 27 80° 24 100° 24 130° 30 angle distance150° 22 feet230° 13 210° 21 180° 18 270° 10 340° 30 330° 18 Table 19.1 Polar Coordinates Contemporary Calculus 4 Graphing Functions in the Polar Coordinate System In the rectangular coordinate system, we have worked with functions given by tables of