# How do you sketch the graph of the polar equation and find the tangents at the pole of r=2cos3theta?

Jan 11, 2017

$\theta = \frac{\pi}{2} , \frac{5}{6} \pi \mathmr{and} \frac{7}{6} \pi$, each twice, for the six tangents. See the graph and explanation.

#### Explanation:

graph{(x^2+y^2)^2-8x^3+6x(x^2+y^2)=0 [-5, 5, -2.5, 2.5]}

The period of $\cos 3 \theta = \frac{2 \pi}{3}$.

In one rotation $\theta \in \left[0 , 2 \pi\right]$ that runs through three periods, 3

loops are created.

At the pole,$r = 0 \to \cos 3 \theta = 0 \to 3 \theta = \frac{\pi}{2} , \frac{3}{2} \pi , \frac{5}{2} \pi , \ldots$

$\to \theta = \frac{\pi}{6} , \frac{\pi}{2} , \frac{5}{6} \pi , \frac{7}{6} \pi , \frac{3}{2} \pi \mathmr{and} 11 \frac{\pi}{6} ,$ for the three loops.

Slope of the tangent

$= \frac{r ' \sin \theta + r \cos \theta}{r ' \cos \theta - r \sin \theta}$

#=(-6sin 3theta sin theta +2 cos 3theta cos theta)

$/ \left(- 6 \sin 3 \theta \cos \theta - 2 \cos 3 \theta \sin \theta\right)$

$= \infty , \pm 1$, at r = 0, against

$\theta = \frac{\pi}{2} , \frac{5}{6} \pi \mathmr{and} \frac{7}{6} \pi$, the tangent taking the same

direction twice.