7.3 Polar Coordinates - Calculus Volume 2 | OpenStax

Defining Polar Coordinates

To find the coordinates of a point in the polar coordinate system, consider Figure 7.27. The point PP has Cartesian coordinates (x,y).(x,y). The line segment connecting the origin to the point PP measures the distance from the origin to PP and has length r.r. The angle between the positive xx-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 rr 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: rr and θ.θ.

Figure 7.27 An arbitrary point in the Cartesian plane.

Using right-triangle trigonometry, the following equations are true for the point P:P:

cosθ=xrsox=rcosθcosθ=xrsox=rcosθ sinθ=yrsoy=rsinθ.sinθ=yrsoy=rsinθ.

Furthermore,

r2=x2+y2andtanθ=yx.r2=x2+y2andtanθ=yx.

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 tanθ=y/xtanθ=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 00 and 2π2π 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 r2=x2+y2.r2=x2+y2.

Converting Points between Coordinate Systems

Given a point PP in the plane with Cartesian coordinates (x,y)(x,y) and polar coordinates (r,θ),(r,θ), the following conversion formulas hold true:

x=rcosθandy=rsinθ,x=rcosθandy=rsinθ, (7.7) r2=x2+y2andtanθ=yx.r2=x2+y2andtanθ=yx. (7.8)

These formulas can be used to convert from rectangular to polar or from polar to rectangular coordinates.

Converting between Rectangular and Polar Coordinates

Convert each of the following points into polar coordinates.

1. (1,1)(1,1)
2. (−3,4)(−3,4)
3. (0,3)(0,3)
4. (53,−5)(53,−5)

Convert each of the following points into rectangular coordinates.

5. (3,π/3)(3,π/3)
6. (2,3π/2)(2,3π/2)
7. (6,−5π/6)(6,−5π/6)

Solution

1. Use x=1x=1 and y=1y=1 in Equation 7.8: r2=x2+y2=12+12r=2andtanθ=yx=11=1θ=π4.r2=x2+y2=12+12r=2andtanθ=yx=11=1θ=π4. Therefore this point can be represented as (2,π4)(2,π4) in polar coordinates.
2. Use x=−3x=−3 and y=4y=4 in Equation 7.8: r2=x2+y2=(−3)2+(4)2r=5andtanθ=yx=−43θ=π-arctan(43)≈2.21.r2=x2+y2=(−3)2+(4)2r=5andtanθ=yx=−43θ=π-arctan(43)≈2.21. The point (-3, 4)(-3, 4) lies in Quadrant IIII. Subtract the value of the reference angle, arctan43arctan43, from ππ to find the radian measure of θθ. Therefore this point can be represented as (5,2.21)(5,2.21) in polar coordinates.
3. Use x=0x=0 and y=3y=3 in Equation 7.8: r2=x2+y2=(3)2+(0)2=9+0r=3andtanθ=yx=30.r2=x2+y2=(3)2+(0)2=9+0r=3andtanθ=yx=30. Direct application of the second equation leads to division by zero. Graphing the point (0,3)(0,3) on the rectangular coordinate system reveals that the point is located on the positive y-axis. The angle between the positive x-axis and the positive y-axis is π2.π2. Therefore this point can be represented as (3,π2)(3,π2) in polar coordinates.
4. Use x=53x=53 and y=−5y=−5 in Equation 7.8: r2=x2+y2=(53)2+(−5)2=75+25r=10andtanθ=yx=−553=−33θ=−π6.r2=x2+y2=(53)2+(−5)2=75+25r=10andtanθ=yx=−553=−33θ=−π6. Therefore this point can be represented as (10,−π6)(10,−π6) in polar coordinates.
5. Use r=3r=3 and θ=π3θ=π3 in Equation 7.7: x=rcosθ=3cos(π3)=3(12)=32andy=rsinθ=3sin(π3)=3(32)=332.x=rcosθ=3cos(π3)=3(12)=32andy=rsinθ=3sin(π3)=3(32)=332. Therefore this point can be represented as (32,332)(32,332) in rectangular coordinates.
6. Use r=2r=2 and θ=3π2θ=3π2 in Equation 7.7: x=rcosθ=2cos(3π2)=2(0)=0andy=rsinθ=2sin(3π2)=2(−1)=−2.x=rcosθ=2cos(3π2)=2(0)=0andy=rsinθ=2sin(3π2)=2(−1)=−2. Therefore this point can be represented as (0,−2)(0,−2) in rectangular coordinates.
7. Use r=6r=6 and θ=−5π6θ=−5π6 in Equation 7.7: x=rcosθ=6cos(−5π6)=6(−32)=−33andy=rsinθ=6sin(−5π6)=6(−12)=−3.x=rcosθ=6cos(−5π6)=6(−32)=−33andy=rsinθ=6sin(−5π6)=6(−12)=−3. Therefore this point can be represented as (−33,−3)(−33,−3) in rectangular coordinates.

Convert (−8,−8)(−8,−8) into polar coordinates and (4,2π3)(4,2π3) into rectangular coordinates.

The polar representation of a point is not unique. For example, the polar coordinates (2,π3)(2,π3) and (2,7π3)(2,7π3) both represent the point (1,3)(1,3) in the rectangular system. Also, the value of rr can be negative. Therefore, the point with polar coordinates (−2,4π3)(−2,4π3) also represents the point (1,3)(1,3) in the rectangular system, as we can see by using Equation 7.8:

x=rcosθ=−2cos(4π3)=−2(−12)=1andy=rsinθ=−2sin(4π3)=−2(−32)=3.x=rcosθ=−2cos(4π3)=−2(−12)=1andy=rsinθ=−2sin(4π3)=−2(−32)=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, rr is the directed distance that the point lies from the origin, and θθ measures the angle that the line segment from the origin to the point makes with the positive xx-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.

Figure 7.28 The polar coordinate system.

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=2r=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 rr 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.

Plotting Points in the Polar Plane

Plot each of the following points on the polar plane.

1. (2,π4)(2,π4)
2. (−3,2π3)(−3,2π3)
3. (4,5π4)(4,5π4)

Solution

The three points are plotted in the following figure.

Figure 7.29 Three points plotted in the polar coordinate system.

Plot (4,5π3)(4,5π3) and (−3,−7π2)(−3,−7π2) on the polar plane.