James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 10.3 Polar Coordinates 659
SOLUtion The points are plotted in Figure 3. In part (d) the point s23, 3y4d is located
three units from the pole in the fourth quadrant because the angle 3y4 is in the second
quadrant and r − 23 is negative.
5π
4
”1, 5π ’
4
O
(2, 3π)
3π
O
O
2π
_ 3
O
3π
4
FIGURE 3
2π ”2, _ ’
3
”_3, 3π ’
4
n
In the Cartesian coordinate system every point has only one representation, but in
the polar coordinate system each point has many representations. For instance, the point
s1, 5y4d in Example 1(a) could be written as s1, 23y4d or s1, 13y4d or s21, y4d.
(See Figure 4.)
5π
4
O
O
_ 3π 4
13π
4
O
O
π
4
”1, 5π 4 ’
3π
”1, _ ’ 4
13π
”1, ’ 4
”_1, π 4 ’
FIGURE 4
In fact, since a complete counterclockwise rotation is given by an angle 2, the point
rep resented by polar coordinates sr, d is also represented by
sr, 1 2nd and s2r, 1 s2n 1 1dd
y
r
P(r, ¨)=P(x, y)
y
where n is any integer.
The connection between polar and Cartesian coordinates can be seen from Figure 5,
in which the pole corresponds to the origin and the polar axis coincides with the positive
x-axis. If the point P has Cartesian coordinates sx, yd and polar coordinates sr, d, then,
from the figure, we have
¨
cos − x r
sin − y r
O
FIGURE 5
x
x
and so
1
x − r cos
y − r sin
Although Equations 1 were deduced from Figure 5, which illustrates the case where
r . 0 and 0 , , y2, these equations are valid for all values of r and . (See the general
definition of sin and cos in Appendix D.)
Equations 1 allow us to find the Cartesian coordinates of a point when the polar coordinates
are known. To find r and when x and y are known, we use the equations
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