Download Polar Coordinates in Mathematics: A Comprehensive Guide and more Study notes Pre-Calculus in PDF only on Docsity!
Jim Lambers MAT 169 Fall Semester 2009- Lecture 33 Notes
These notes correspond to Section 9.3 in the text.
Polar Coordinates
Throughout this course, we have denoted a point in the plane by an ordered pair (푥, 푦), where the numbers 푥 and 푦 denote the directed (i.e., signed positive or negative) distance between the point and each of two perpendicular lines, the 푥-axis and the 푦-axis. The elements of this ordered pair are called coordinates, and the coordinates used in this particular method of identifying points in the plane are called Cartesian coordinates. In this lecture, we introduce an alternative coordinate system known as the polar coordinate system. In this system, a point in the plane is identified by an ordered pair (푟, 휃), where:
∙ 푟 is the directed distance from a point designated as the pole, and
∙ 휃 is the angle, in radians, that a ray between the pole and the point makes with a ray designated as the polar axis.
The coordinates 푟 and 휃 are called polar coodinates. The pole is the point (0, 0) in Cartesian coordinates, and has polar coordinates (0, 휃) for any value of 휃. The polar axis corresponds to the positive 푥-axis. An angle 휃 is considered positive if measured in the counterclockwise direction from the polar axis, and negative if measured in the clockwise direction. Using these conventions, the Cartesian coordinates of a point can easily be obtained from the polar coordinates using the relations
푥 = 푟 cos 휃, 푦 = 푟 sin 휃.
Since sin 휃 and cos 휃 are not one-to-one, and since 푟 is allowed to assume negative values, it follows that each point in the plane has infinitely many representations in polar coordinates.
Example Compute the Cartesian coordinates of the following points whose polar coordinates are given.
- (1, 휋/4)
- (− 1 , 5 휋/4)
- (1, 9 휋/4)
Solution Using the relations 푥 = 푟 cos 휃, 푦 = 푟 sin 휃,
we have:
- 푥 = 1 ⋅ cos(휋/4) =
2 /2, 푦 = 1 ⋅ sin(휋/4) =
- 푥 = − cos(5휋/4) = −(−
2 /2, 푦 = − sin(휋/4) = −(−
- 푥 = 1 ⋅ cos(9휋/4) = cos(휋/4) =
2 /2, 푦 = 1 ⋅ sin(9휋/4) = sin(휋/4) =
The polar coordinates of a point can be obtained from the Cartesian coordinates as follows:
푟 = 푥^2 + 푦^2 , tan 휃 =
It should be noted that because tan 휃 is not one-to-one on the interval 0 ≤ 휃 < 2 휋, it is necessary to consider the signs of 푥 and 푦 in order to make sure that the proper value of 휃 is used to represent the point (푥, 푦). Otherwise, the point (푟, 휃) may lie in the wrong quadrant of the plane.
Example Compute the polar coordinates of the following points whose Cartesian coordinates are given.
- (−
Solution Using the relations 푟^2 = 푥^2 + 푦^2 , tan 휃 =
we have:
푟^2 = (−
3 /2)^2 + (1/2)^2 = 3/4 + 1/4 = 1, tan 휃 = −
It follows that 푟 = 1. Because the 푥-coordinate of the point is negative, we should seek a value of 휃 that lies in the interval (휋/ 2 , 3 휋/2). However, the range of the inverse tangent function lies in the interval (−휋/ 2 , 휋/2), and therefore 휃 = tan−^1 (− 1 /
- = −휋/6. Since tangent has a period of 휋, it follows that
tan(휃 + 휋) = tan 휃
for any 휃; in other words, its values repeat after every 휋 units. Since
tan(5휋/6) = tan(−휋/6 + 휋) = tan(−휋/6) = −
Figure 1: Curve described by the polar equation 푟 = cos 2휃, where 0 ≤ 휃 ≤ 2 휋.
We now determine the slope of a tangent line of a polar curve. If the curve can be described by an equation of the form 푦 = 퐹 (푥) for some differentiable function 퐹 , then, by the Chain Rule,
푑푦 푑휃
but since 퐹 ′(푥) = 푑푦/푑푥, it follows that
푑푦 푑푥
Expressing 푥 and 푦 in polar coordinates and applying the Product Rule yields
푑푦 푑푥
푑푦 푑휃 푑푥 푑휃
푑푟 푑휃 sin^ 휃^ +^ 푟^ cos^ 휃 푑푟 푑휃 cos^ 휃^ −^ 푟^ sin^ 휃
It can be shown that this result also holds for curves that cannot be described by an equation of the form 푦 = 퐹 (푥).
Figure 2: Curve described by the polar equation 푟 = sin 휃, where 0 ≤ 휃 ≤ 2 휋.
We make the following observations about tangents to polar curves, based on the above expres- sion for their slope:
∙ Horizontal tangents occur when 푑푦/푑휃 = 0, provided that 푑푥/푑휃 ∕= 0.
∙ Vertical tangents occur when 푑푥/푑휃 = 0, provided that 푑푦/푑휃 ∕= 0.
∙ At the pole, when 푟 = 0, the slope of the tangent is given by
푑푦 푑푥
푑푟 푑휃 sin^ 휃 푑푟 푑휃 cos^ 휃^
= tan 휃
provided 푑푟/푑휃 ∕= 0.
Example Given the curve defined by the polar equation 푟 = sin 휃, where 0 ≤ 휃 ≤ 휋, determine the values of 휃 at which the tangent to the curve is either horizontal or vertical.
Summary
∙ A point can be represented by polar coordinates (푟, 휃), where 푟 is the distance between the point and the origin, or pole, and 휃 is the angle that a line segment from the pole to the point makes with the positive 푥-axis.
∙ To convert from polar coordinates to Cartesian coordinates (푥, 푦), one can use the formulas 푥 = 푟 cos 휃 and 푦 = 푟 sin 휃.
∙ To convert from Cartesian coordinates to polar coordinates, one can use 푟 =
p 푥^2 + 푦^2 , and 휃 = tan−^1 (푦/푥) if 푥 < 0. If 푥 < 0, then 휃 = tan−^1 (푦/푥) + 휋. If 푥 = 0, 휃 = 휋/2 if 푦 > 0, and −휋/2 if 푦 < 0.
∙ To graph a curve defined by a polar equation of the form 푟 = 푓 (휃), one can compute 푟 for various values of 휃, and use polar coordinates to plot the corresponding points on the curve.
∙ To compute the slope of the tangent to a polar curve 푟 = 푓 (휃), one can differentiate 푥 = 푓 (휃) cos 휃 and 푦 = 푓 (휃) sin 휃 with respect to 휃, and then use the relation 푑푦/푑푥 = (푑푦/푑휃)/(푑푥/푑휃).
