In polar coordinates, for $(r,\theta)$ given point, we know that $r$ defined on $r\geq 0$ and $0 \leq \theta \leq 2\pi $. Because $r$ shows the distance between the point and the origin. My question about graph of $r=\sin (2 \theta)$. For example if $\theta = 3\pi /4$ then $r=-1$ undefined. But in wolfram alpha, the equation is defined for $\theta = 3\pi /4$. Moreover, for $$ \pi /2 \leq \theta < \pi , \qquad \sin 2\theta <0 \qquad \text{and} \qquad r<0$$ Hence $r$ is undefined. But, wolfram can draw the graph.
Where is the miskate?
Best Answer
With polar coordinates $(r,\theta) $, we have cartesians by
$$x=r\cos (\theta) $$ and $$y=r\sin (\theta) $$
so $r $ can take negative values.
for example, if $\theta=\frac {3\pi}{4} $, $r=-1$ and this gives the point
$$(x,y)=(\frac {\sqrt {2}}{2},-\frac {\sqrt {2}}{2}) .$$