How do you sketch the graph of the polar equation and find the tangents at the pole of #r=3sin2theta#?

1 Answer
Feb 8, 2017

#theta= 0 and 3/2pi#, for anticlockwise tracing of the #Q_1# loop and .#theta = pi and pi/2#, for the #Q_3# loop,

Explanation:

The period for the graph is #(2pi)/2=pi#.

As #r = sqrt(x^2+y^2)>=0#, #2theta in Q_1 or Q_2#, setting

#theta in Q_1#. Perhaps, I am one of a few in the teaching

community to disallow r-negative loops that appear in

#Q_4 and Q_2#..

In my count, there is just one loop and that is in #Q_1#, for the

period #theta in [0, pi]#. For the next period theta in #[pi, 2pi]#, the

second r-positive loop in #Q_3# is created

In respect of the first loop, the tangency is either through #theta

=0# or in the perpendicular direction #theta=3/2pi#, for

anticlockwise tracing. For the second, these angles are #pi and

pi/2#.

Note: The Socratic utility adheres to r >=0 logic. There might be

some graphic devices that create four loops for this graph, and 4n

loops, for #r = a sin 2ntheta#, n = 2, 3, ....

graph{((x^2+y^2)^1.5-6xy)=0 [-5, 5, -2.5, 2.5]}