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

Section 10.3 Polar Coordinates 663
Symmetry
When we sketch polar curves it is sometimes helpful to take advantage of symmetry. The
following three rules are explained by Figure 14.
(a) If a polar equation is unchanged when is replaced by 2, the curve is sym metric
about the polar axis.
(b) If the equation is unchanged when r is replaced by 2r, or when is replaced by
1 , the curve is symmetric about the pole. (This means that the curve remains
unchanged if we rotate it through 180° about the origin.)
(c) If the equation is unchanged when is replaced by 2 , the curve is sym metric
about the vertical line − y2.
(r, ¨)
(r, π-¨)
(r, ¨)
O
¨
_¨
(_r, ¨)
O
(r, ¨)
π-¨
O
¨
(r, _¨)
(a) (b) (c)
FIGURE 14
The curves sketched in Examples 6 and 8 are symmetric about the polar axis, since
coss2d − cos . The curves in Examples 7 and 8 are symmetric about − y2 because
sins 2 d − sin and cos 2s 2 d − cos 2. 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 0 < < y2 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 r − f sd, we regard as a parameter and write its
parametric equations as
x − r cos − f sd cos
y − r sin − f sd sin
Then, using the method for finding slopes of parametric curves (Equation 10.2.1) and the
Product Rule, we have
3
dy
dy
dx − d
−
dx
d
dr
sin 1 r cos
d
dr
d
cos 2 r sin
We locate horizontal tangents by finding the points where dyyd − 0 (provided that
dxyd ± 0). Likewise, we locate vertical tangents at the points where dxyd − 0 (provided
that dyyd ± 0).
Notice that if we are looking for tangent lines at the pole, then r − 0 and Equation 3
simplifies to
dy
dx − tan
dr
if
d ± 0
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