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

Feb 8, 2017

$\theta = 0 \mathmr{and} \frac{3}{2} \pi$, for anticlockwise tracing of the ${Q}_{1}$ loop and .$\theta = \pi \mathmr{and} \frac{\pi}{2}$, for the ${Q}_{3}$ loop,

#### Explanation:

The period for the graph is $\frac{2 \pi}{2} = \pi$.

As $r = \sqrt{{x}^{2} + {y}^{2}} \ge 0$, $2 \theta \in {Q}_{1} \mathmr{and} {Q}_{2}$, setting

$\theta \in {Q}_{1}$. Perhaps, I am one of a few in the teaching

community to disallow r-negative loops that appear in

${Q}_{4} \mathmr{and} {Q}_{2}$..

In my count, there is just one loop and that is in ${Q}_{1}$, for the

period $\theta \in \left[0 , \pi\right]$. For the next period theta in $\left[\pi , 2 \pi\right]$, the

second r-positive loop in ${Q}_{3}$ is created

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

=0$\mathmr{and} \in t h e p e r p e n \mathrm{di} c \underline{a} r \mathrm{di} r e c t i o n$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 $r = a \sin 2 n \theta$, n = 2, 3, ....

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