10.6 Polar Coordinates:
So far, you have been representing graphs of equations as collections of points (x, y) on the rectangular coordinate system, where x and y represent the directed distances from teh coordinate axes to the point (x, y). In this section, you will study a second system - the Polar Coordinate System.
I. To form the polar coordinate system in the plane, fix a point O, called the pole (or origin) and construct from O an initial ray called the polar axis. Then each point P in the plane can be assigned polar coodinates (r,q ) as follows:
1. r = directed distance from O to P.
2. q = directed angle, counterclockwise from polar axis to segment OP.
To find other points in the same place, do the following:
(r, q ) = (r, q ± 2np )
or
(r, q) = (-r, q ±(2n + 1)p )
Example: Given the polar coordinate (3, p/4)
= (3, p/4 + 2p) = (3, 9p/4)
OR
(-3, p/4 + (2 (0) +1)p) = (-3, 5p/4)
(-3, p/4 - p) = (-3, -3p/4)
II. Coordinate Conversion:
Because (x, y) lies on a circle of radius r, it follows that r2 = x2 + y2. Moreover, for r> 0, the definitions of the trigonometric functions imply:
tan q = y / x
cos q = x/ r
sin q = y/r
The Polar Coordinates (r, q) are related to the rectangular coordinates (x, y) as follows:
x = r cos q
y = r sin q
tan q = y/x
r2 = x2 + y2
III. Polar to Rectangular Conversion
Example 2: Convert each point to rectangular coordinates
a). (-2, 7p/6) means r = -2 and q = 7p/6
x = (-2)(cos 7p/6) = (-2)(- Ö(3)/2) = Ö3
y = (-2)(sin 7p/6) = (-2)(-1/2) = 1
Therefore the rectangular coordinate is (Ö3 , 1)
b). (8.25, 3.5) means r = 8.25 and q = 3.5 radians
x = (8.25)(cos 3.5) = -7.72576767
y = (8.25)(sin 3.5) = -2.89391628
Therefore the rectangular coordinate is (-7.72576767, -2.89391628)
IV. Rectangle to Polar Conversion
Example 3: Convert each point to polar coordinates
a) Let 0£ q £ 2p and using the rectangular coordinate (0, -5)
tan q = -5/0 = undefined, therefore q = 270° based on where the point is.
r = Ö (x2 + y2)
= Ö(02 + (-5)2)
= 5
Therefore a polar coordinate is (5, 270°) or (5, 3p/2)
and the negative would be (-5, -90°) or (-5, p/2)
b) (3, -1)
Tan q = -1/3
q = -18.43494882° or -.3217505544 radians
so q = 341.5650512° or 5.961434753 radians
r = Ö(32 + (-1)2) = Ö(9 + 1) = Ö10
Therefore the polar coordinate is (Ö10, 5.961434753 radians)
To find another coordinate for the same location: 5.961434753-p = 2.819842099
so (-Ö10 , 2.819842099 radians)
V. Equation Convertion
Recall x = r cos q and y = r sin q and x2 + y2 = r2
Given: x2 + y2 - 6x = 0 substitute in the above equations for x and y
r2 - 6(r cos q) = 0
r ( r - 6 cos q) = 0
r = 0 and r - 6 cos q = 0
r = 6 cos q
so the polar equation is r = 6 cos q
VI. Convert the polar equation to rectangular form:
Example: r = 4 cos q
r2 = 4r cos q
x2 + y2 = 4x
x2 -4x + y2 = 0 would be the rectangular equation
Example: q = 5p/3
Tan q = tan 5p/3 = -Ö3
y/x = -Ö3/1
y = -Ö3 x
y = xÖ3 = 0
Example: r = 6/(2cosq - 3 sin q)
r = 6 / (2(x/r) - 3(y/r))
r = 6r/(2x - 3y)
2x - 3y = 6r/r
2x - 3y = 6
