find the equations of the tangents at the pole.

Question

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?

Answer 1

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 $\dfrac{-\pi}{6} = \dfrac{11\pi}{6}$.

Answer 2

$$\sin{3\theta}=-1=\sin{{3\pi\over 2}}$$

And this means

$$3\theta={3\pi\over 2}+2k\cdot\pi$$

Where $k\in\Bbb{Z}$. Therefore

$$\theta={\pi\over 2}+{2k\pi\over 3}$$

For $k=0$ we get $\pi/2$.

For $k=1$ we get $7\pi/6\equiv -5\pi/6\pmod{2\pi}$

For $k=2$ we get $11\pi/6\equiv -\pi/6\pmod{2\pi}$