Question

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

Answer 1

`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]}