The curves sketched in examples 6 and 8 are symmetric

The curves sketched in Examples 6 and 8 are symmetric about the polar axis, since
. The curves in Examples 7 and 8 are symmetric about because
and . The four-leaved rose is also symmetric
about the pole. These symmetry properties could have been used in sketching the curves.
For instance, in Example 6 we need only have plotted points for and then
reflected about the polar axis to obtain the complete circle.
Tangents to Polar Curves
To find a tangent line to a polar curve , we regard as a parameter and write its
parametric equations as
Then, using the method for finding slopes of parametric curves (Equation 10.2.1) and the
Product Rule, we have
We locate horizontal tangents by finding the points where (provided that
). Likewise, we locate vertical tangents at the points where (pro-
vided that ).
Notice that if we are looking for tangent lines at the pole, then and Equation 3 sim-
plifies to
r r
2
O
(r, ¨)
(_r, ¨)
O
(r, ¨)
(r, _¨)
_¨
¨
(a) (b) (c)
FIGURE 14
O
(r, ¨) (r, π-¨)
π-¨
¨
2 cos cos
cos 2 cos 2 sin sin
0 2
r f
y r sin f sin x r cos f cos
dy
dx
dy
d
dx
d
dr
d
sin r cos
dr
d
cos r sin
3
dy d 0
dx d 0 dx d 0
dy d 0
r 0
dr
d
0 if
dy
dx
tan

684
CHAPTER 10
For instance, in Example 8 we found that when or . This
means that the lines and (or and ) are tangent lines to
at the origin.
(a) For the cardioid of Example 7, find the slope of the tangent line
when
(b) Find the points on the cardioid where the tangent line is horizontal or vertical.
r cos 2 0 4 3 4
4 3 4 y x y x
r cos 2
EXAMPLE 9
.
and
are
cos
sin
2
,
,
1
2
5π
π
π
2
¨

POLAR COORDINATES
685
NOTE
Instead of having to remember Equation 3, we could employ the method used to
derive it. For instance, in Example 9 we could have written
Then we have
which is equivalent to our previous expression.