Section  10.3     MATH  1152Q  Part  2   Polar  Coordinates   Notes  

Page 1 of 3

Polar Curves The graph of a polar equation consists of all points that have at least one polar

representation whose coordinates satisfy the equation.

Cardioid Four-leaved Rose

( )r f q= P

( ),r q

4r =6pq =

4cosr q= 4sinr q=

2 2sinr q= + 4cos2r q=

Section  10.3     MATH  1152Q  Part  2   Polar  Coordinates   Notes  

Page 2 of 3

Symmetry When we sketch polar curves, it is sometimes helpful to take advantage of symmetry.

•   The polar curve is symmetric about the polar axis if Ø   and are both on the curve OR

Ø   and are both on the curve.

•   The polar curve is symmetric about the vertical line if

Ø   and are both on the curve OR

Ø   and are both on the curve. •   The polar curve is symmetric about the pole if

Ø   and are both on the curve OR

Ø   and are both on the curve. Tangents to Polar Curve 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 and the Product Rule, we have Notice that if we are looking for tangent lines at the pole, then . Thus the above equation simplifies to

( ),r q ( ),r q-

( ),r q ( ),r q p- - +

2p

q =

( ),r q ( ),r q- -

( ),r q ( ),r q p- +

( ),r q ( ),r q-

( ),r q ( ),r q p+

( )r f q= q

cossin

x ry r

qq

=ìí =î

Þ ( )( )cossin

x fy f

q qq q

=ìïí =ïî

0r =

Section  10.3     MATH  1152Q  Part  2   Polar  Coordinates   Notes  

Page 3 of 3

Example:

Find the slope of the tangent line of the cardioid when .

Write an equation of this tangent line.

1 sinr q= +3pq =