The rectangular coordinate system (or Cartesian plane) provides a means of mapping points to ordered pairs and ordered pairs to points. This is called a one-to-one mapping from points in the plane to ordered pairs. The polar coordinate system provides an alternative method of mapping points to ordered pairs. In this section we see that in some circumstances, polar coordinates can be more useful than rectangular coordinates.
To find the coordinates of a point in the polar coordinate system, consider Figure 7.27. The point P P has Cartesian coordinates ( x , y ) . ( x , y ) . The line segment connecting the origin to the point P P measures the distance from the origin to P P and has length r . r . The angle between the positive x x -axis and the line segment has measure θ . θ . This observation suggests a natural correspondence between the coordinate pair ( x , y ) ( x , y ) and the values r r and θ . θ . This correspondence is the basis of the polar coordinate system . Note that every point in the Cartesian plane has two values (hence the term ordered pair) associated with it. In the polar coordinate system, each point also has two values associated with it: r r and θ . θ .
Using right-triangle trigonometry, the following equations are true for the point P : P :
cos θ = x r so x = r cos θ cos θ = x r so x = r cos θ sin θ = y r so y = r sin θ . sin θ = y r so y = r sin θ . r 2 = x 2 + y 2 and tan θ = y x . r 2 = x 2 + y 2 and tan θ = y x .Each point ( x , y ) ( x , y ) in the Cartesian coordinate system can therefore be represented as an ordered pair ( r , θ ) ( r , θ ) in the polar coordinate system. The first coordinate is called the radial coordinate and the second coordinate is called the angular coordinate . Every point in the plane can be represented in this form.
Note that the equation tan θ = y / x tan θ = y / x has an infinite number of solutions for any ordered pair ( x , y ) . ( x , y ) . However, if we restrict the solutions to values between 0 0 and 2 π 2 π then we can assign a unique solution to the quadrant in which the original point ( x , y ) ( x , y ) is located. Then the corresponding value of r is positive, so r 2 = x 2 + y 2 . r 2 = x 2 + y 2 .
Given a point P P in the plane with Cartesian coordinates ( x , y ) ( x , y ) and polar coordinates ( r , θ ) , ( r , θ ) , the following conversion formulas hold true:
x = r cos θ and y = r sin θ , x = r cos θ and y = r sin θ , r 2 = x 2 + y 2 and tan θ = y x . r 2 = x 2 + y 2 and tan θ = y x .These formulas can be used to convert from rectangular to polar or from polar to rectangular coordinates.
Convert each of the following points into polar coordinates.
Convert each of the following points into rectangular coordinates.
r 2 = x 2 + y 2 = 1 2 + 1 2 r = 2 and tan θ = y x = 1 1 = 1 θ = π 4 . r 2 = x 2 + y 2 = 1 2 + 1 2 r = 2 and tan θ = y x = 1 1 = 1 θ = π 4 .
r 2 = x 2 + y 2 = ( −3 ) 2 + ( 4 ) 2 r = 5 and tan θ = y x = − 4 3 θ = π - arctan ( 4 3 ) ≈ 2 . 21 . r 2 = x 2 + y 2 = ( −3 ) 2 + ( 4 ) 2 r = 5 and tan θ = y x = − 4 3 θ = π - arctan ( 4 3 ) ≈ 2 . 21 .
r 2 = x 2 + y 2 = ( 3 ) 2 + ( 0 ) 2 = 9 + 0 r = 3 and tan θ = y x = 3 0 . r 2 = x 2 + y 2 = ( 3 ) 2 + ( 0 ) 2 = 9 + 0 r = 3 and tan θ = y x = 3 0 .
r 2 = x 2 + y 2 = ( 5 3 ) 2 + ( −5 ) 2 = 75 + 25 r = 10 and tan θ = y x = −5 5 3 = − 3 3 θ = − π 6 . r 2 = x 2 + y 2 = ( 5 3 ) 2 + ( −5 ) 2 = 75 + 25 r = 10 and tan θ = y x = −5 5 3 = − 3 3 θ = − π 6 .
x = r cos θ = 3 cos ( π 3 ) = 3 ( 1 2 ) = 3 2 and y = r sin θ = 3 sin ( π 3 ) = 3 ( 3 2 ) = 3 3 2 . x = r cos θ = 3 cos ( π 3 ) = 3 ( 1 2 ) = 3 2 and y = r sin θ = 3 sin ( π 3 ) = 3 ( 3 2 ) = 3 3 2 .
x = r cos θ = 2 cos ( 3 π 2 ) = 2 ( 0 ) = 0 and y = r sin θ = 2 sin ( 3 π 2 ) = 2 ( −1 ) = −2. x = r cos θ = 2 cos ( 3 π 2 ) = 2 ( 0 ) = 0 and y = r sin θ = 2 sin ( 3 π 2 ) = 2 ( −1 ) = −2.
x = r cos θ = 6 cos ( − 5 π 6 ) = 6 ( − 3 2 ) = −3 3 and y = r sin θ = 6 sin ( − 5 π 6 ) = 6 ( − 1 2 ) = −3. x = r cos θ = 6 cos ( − 5 π 6 ) = 6 ( − 3 2 ) = −3 3 and y = r sin θ = 6 sin ( − 5 π 6 ) = 6 ( − 1 2 ) = −3.
Convert ( −8 , −8 ) ( −8 , −8 ) into polar coordinates and ( 4 , 2 π 3 ) ( 4 , 2 π 3 ) into rectangular coordinates.
The polar representation of a point is not unique. For example, the polar coordinates ( 2 , π 3 ) ( 2 , π 3 ) and ( 2 , 7 π 3 ) ( 2 , 7 π 3 ) both represent the point ( 1 , 3 ) ( 1 , 3 ) in the rectangular system. Also, the value of r r can be negative. Therefore, the point with polar coordinates ( −2 , 4 π 3 ) ( −2 , 4 π 3 ) also represents the point ( 1 , 3 ) ( 1 , 3 ) in the rectangular system, as we can see by using Equation 7.8:
x = r cos θ = −2 cos ( 4 π 3 ) = −2 ( − 1 2 ) = 1 and y = r sin θ = −2 sin ( 4 π 3 ) = −2 ( − 3 2 ) = 3 . x = r cos θ = −2 cos ( 4 π 3 ) = −2 ( − 1 2 ) = 1 and y = r sin θ = −2 sin ( 4 π 3 ) = −2 ( − 3 2 ) = 3 .
Every point in the plane has an infinite number of representations in polar coordinates. However, each point in the plane has only one representation in the rectangular coordinate system.
Note that the polar representation of a point in the plane also has a visual interpretation. In particular, r r is the directed distance that the point lies from the origin, and θ θ measures the angle that the line segment from the origin to the point makes with the positive x x -axis. Positive angles are measured in a counterclockwise direction and negative angles are measured in a clockwise direction. The polar coordinate system appears in the following figure.
The line segment starting from the center of the graph going to the right (called the positive x-axis in the Cartesian system) is the polar axis . The center point is the pole , or origin, of the coordinate system, and corresponds to r = 0 . r = 0 . The innermost circle shown in Figure 7.28 contains all points a distance of 1 unit from the pole, and is represented by the equation r = 1 . r = 1 . Then r = 2 r = 2 is the set of points 2 units from the pole, and so on. The line segments emanating from the pole correspond to fixed angles. To plot a point in the polar coordinate system, start with the angle. If the angle is positive, then measure the angle from the polar axis in a counterclockwise direction. If it is negative, then measure it clockwise. If the value of r r is positive, move that distance along the terminal ray of the angle. If it is negative, move along the ray that is opposite the terminal ray of the given angle.
Plot each of the following points on the polar plane.
