Q:

find the equations of the tangents at the pole.

For the graph with polar equation $r = 1 + sin 3\theta$, find the equations of the tangents at the pole.
My attempt, 
When $r=0$, 
$\sin3 \theta=-1$
$\theta=\frac{\pi}{2}, \frac{7\pi}{2},\frac{11\pi}{2}$
But the given answer is $\frac{-5\pi}{6}, \frac{-\pi}{6},\frac{\pi}{2}$
Why?

A:

Notice that $\sin \dfrac{7\pi}{2}$ = -1. $\therefore 3\theta = \dfrac{7\pi}{2}$... It's a similar mistake for your answer $\dfrac{11\pi}{2}$ 
So you've got an arithmetic error, and that's basically it. Just divide each of your answers (except for $\dfrac{\pi}{2}$) by 3, and you'll get the correct answer, because $\dfrac{-5\pi}{6}$ = $\dfrac{7\pi}{6}$ and $\dfrac{-\pi}{6} = \dfrac{11\pi}{6}$.

