Definition (Polar Coordinates) Figure: The polar coordinates (r, )ofapointp

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1 Polar Coordinates Acoordinatesystemusesapairofnumberstorepresentapointontheplane. We are familiar with the Cartesian or rectangular coordinate system, (x, y). It is not always the most convenient system to use depending on the nature of our problem. The polar coordinate system, (r, ), is convenient if we want to consider radial distance from a fixed point (origin, or pole) and bearing (direction). Figure: The polar coordinates (r, )ofapointp Definition (Polar Coordinates) Choose a point O on the plane and call it the pole (or origin). Call the horizontal ray drawn from the pole to the right the polar axis, which coincides with the positive x-axis of the rectangular coordinate system. For any point P on the plane, let be the signed (radian) measure of an angle with the polar ray as initial ray and ± OP! as terminal ray, and let r be the signed distance from O to P determined by the orientation of the terminal ray. In the ordered pair (r, ), r and are called the polar coordinates of P. Math 267 (University of Calgary) Fall 2015, Winter / 19

2 Each point P corresponds to infinitely many values of, counter-clockwisemeasuresbeingpositive. If we choose OP!! as the terminal ray, r 0. If we choose OP as the terminal ray, r apple 0. Thus, in the polar coordinate system, the ordered pairs (2, 2 ), ( point. 2, )and(2, 0) all represent the same Figure: Three di erent polar coordinates representations of the same point P (0, )representsthepoleforanyvalueof. Math 267 (University of Calgary) Fall 2015, Winter / 19

3 Example If a point P has polar coordinates 1,,findallpossibleorderedpairsofpolarcoordinatesforP. 3 Solution: The point P is located at a distance of 1 unit from O along the ray which makes a positive (counter-clockwise) angle of 3 with the polar axis. Using positive distance r =1, can be any one of 3 +2n (n =0, ±1, ±2,...) Using negative distance r = 1, can be any one of + +2n (n =0, ±1, ±2,...) 3 The ordered pairs of polar coordinates of P are 1, 3 +2n and 1, n (n =0, ±1, ±2,...) Figure: In general, (r, )isalsorepresentedbypolarcoordinates(r, +2n ) and( r, + +2n ) Math 267 (University of Calgary) Fall 2015, Winter / 19

4 Connection between Polar and Cartesian Coordinates Theorem (Polar and Cartesian Coordinates) Suppose point P has Cartesian coordinates (x, y) andpolarcoordinates(r, ). Then From polar to Cartesian: x = r cos, y = r sin Figure: Conversions between Cartesian and Polar Coordinates From Cartesian to polar: r 2 = x 2 + y 2, tan = y x Math 267 (University of Calgary) Fall 2015, Winter / 19

5 Curves in Polar Coordinates Example 1 The point with polar coordinates 2, 5 has Cartesian coordinates 6 x =2cos 5 p! 3 6 =2 = p 3 2 and Thus, (x, y) = p3, 1. y =2sin =2 = The point with Cartesian coordinates (1, 1) is in the fourth quadrant (Quadrant IV). It has polar coordinates satisfying r 2 =1 2 +( 1) 2 =2 and tan = 1 1 = 1. p2, 7 One possible solution is (r, )=. 4 p2, 3 Another solution is (r, )=. 4 Math 267 (University of Calgary) Fall 2015, Winter / 19

6 Curves in Polar Coordinates We would like to sketch the curve on the plane defined by a polar equation such as r =3 = 4 r =2sin r =cos3 r = The graph of a polar equation consists of all points that have at least one pair of polar coordinates (r, ) satisfying the equation. We do not require all pairs of polar coordinates of the point to satisfy the equation. One pair su ces. Figure: When does a point belong to a polar curve? The pair (1, 1) satisfies r =, but the pair (1, 1+2 ) doesnotsatisfyr =. We would consider the common point they represent as belonging to the curve. Math 267 (University of Calgary) Fall 2015, Winter / 19

7 In the polar equation r =3,r refers to the distance from the pole to the point. r =3isthecirclecenteredatthepolewithradius3. Figure: Polar curve r =3inthexy-plane; Line r =3inther -plane Note that r =3wouldbeaverticallineonther -plane (with horizontal r-axis and vertical -axis). Math 267 (University of Calgary) Fall 2015, Winter / 19

8 In the polar equation =, refers to one of the polar angles of the point. 4 = 4 is the straight line through the pole making an angle of 4 radians with the polar axis. Figure: Polar curve = 4 in the xy-plane; Line = 4 in the r -plane Note that = 4 would be a horizontal line on the r -plane. Math 267 (University of Calgary) Fall 2015, Winter / 19

9 Example Sketch and describe the curve defined by the polar equation r =2sin. Solution: Method 1 Make a table of values. Plot and connect the dots. r=2sin 0 0 /6 p 1 /4 p 2 /3 3 /2 p 0 2 /3 p 3 3 /4 2 5 /6 1 0 Figure: Polar curve r =2sin (in the xy-plane) Math 267 (University of Calgary) Fall 2015, Winter / 19

10 Method 2 Convert the polar equation into a Cartesian equation. Multiply by r: r 2 =2rsin. Using r 2 = x 2 + y 2 and y =2sin : x 2 + y 2 =2y () x 2 + y 2 2y =0 () x 2 + y 2 2y +1=1 () x 2 +(y 1) 2 =1. Circle with centre (0, 1) and radius 1. Figure: r =2sin is the circle centered at (0, 1) with radius 1 in the xy-plane Remark Let Q be the point (0, 2). The line PQ is perpendicular to the line OP. Studying the right-angled triangle OPQ, weseethatthelengthr of OP is 2 sin. Math 267 (University of Calgary) Fall 2015, Winter / 19

11 Example More generally, here are two common polar curves. We will encounter them again in future examples. 1 r =2asin () x 2 +(y a) 2 = a 2,circlewithcentre(0, a) andradius a. 2 r =2acos () (x a) 2 + y 2 = a 2,circlewithcentre(a, 0) and radius a. Figure: r =2a sin and r =2a cos, wherea > 0 Math 267 (University of Calgary) Fall 2015, Winter / 19

12 Example Sketch the curve r =cos3. 1 Solution: The curve r =cos is a circle with centre 2, 0 and radius 1 2. Replace by 3 in the equation: The curve loops around three times as fast. The curve r =cos traces the circle once when increases from 2 to 2. The curve r =cos3 traces a loop once when increases from 6 to 6. Figure: Aplotofr =cos3 using Maple 18 Math 267 (University of Calgary) Fall 2015, Winter / 19

13 Recall: The curve r =cos traces the same circle a second time when increases from 2 to 3 2. In this interval ( in Quadrants II and III), r becomes negative and the curve is traced in Quadrants IV and I. So, it retraces the same circle. Figure: Examine how the circle on the right is retraced when increases from 2 to 3 2 The curve r =cos3 traces the loop a second time when increases from 6 to 2. Like the case with the circle, r is negative. But unlike the case with the circle, this new loop does not coincide with the first loop. The second loop appears in Quadrant III. The completed graph has exactly three loops. Math 267 (University of Calgary) Fall 2015, Winter / 19

14 Further examples/exercises Example Sketch the following curves by (a) making a table of values and plotting, or by (b) using WolframAlpha. 1 r =3sin4 2 r =1+cos 3 r =1+2cos 4 r =sin 9 5 Math 267 (University of Calgary) Fall 2015, Winter / 19

15 Area Recall: The formula for the area of a sector: A = 1 2 r 2 Theorem (Area in Polar Coordinates) Suppose that D is a region on the xy-plane bounded between polar curves as follows: 0 apple r apple f ( ), apple apple, where f is non-negative and continuous. Figure: 0 apple r apple f ( ), apple apple Then the area of D is given by A = 1 2 Z (f ( )) 2 d Math 267 (University of Calgary) Fall 2015, Winter / 19

16 Proof (Outline): Slice D radially into pieces resembling sectors. Figure: 0 apple r apple f ( ), apple apple Each small piece has small angle d and approximate radius r = f ( ). The area of a small piece is approximately 1 2 (f ( ))2 d. The total area is the integral. Math 267 (University of Calgary) Fall 2015, Winter / 19

17 Example Find the area of bounded inside one of the loops of the polar curve r =cos(3 ). Solution: Figure: Aplotofr =cos3 using Maple 18 Consider the loop bounded as increases from 6 to 6. A = 1 2 Z /6 /6 (cos (3 )) 2 d = 1 2 Z /6 /6 1+cos(6 ) 2 d = sin(6 ) /6 2 /6 = 12. Math 267 (University of Calgary) Fall 2015, Winter / 19

18 Theorem (Area in Polar Coordinates) Suppose that D is a region on the xy-plane described by f ( ) apple r apple g( ), apple apple, where f is non-negative and continuous. Figure: 0 apple f ( ) apple r apple g( ), apple apple Then the area of D is given by A = 1 2 Z (g( )) 2 (f ( )) 2 d Math 267 (University of Calgary) Fall 2015, Winter / 19

19 Further examples/exercises Example 1 Find the area of the bounded region enclosed by the given polar curves (considering only r 0). 1 r =, 0 apple apple 2 2 r = p cos, 2 apple apple 2 3 One of the loops of r =sin2 4 r =2 cos 2 Consider the circle r =6sin and the cardioid r =2+2sin. 1 Sketch them. 2 Find their intersection. 3 Identify the bounded region inside the circle and outside the cardioid. Find its area. 4 Identify the bounded region outside the circle and inside the cardioid. Find its area. 5 Find the area of the region inside both the circle and the cardioid. Math 267 (University of Calgary) Fall 2015, Winter / 19