Answer:
Step-by-step explanation:
The given equation is
[tex]r=2+3sin \theta[/tex]
Notice that this equation is in polar form.
If [tex](r, \theta)[/tex] can be replace with [tex](r, \pi - \theta)[/tex] or [tex](-r, -\theta)[/tex], then the graph is symmetric to the line [tex]\theta = \frac{\pi}{2}[/tex], which is a vertical line.
Let's evaluate the given equation.
[tex]r=2+3sin \theta=2+3sin(\pi -\theta)= 2-3sin(- \theta)[/tex]
But, [tex]sin(-\theta)=-sin \theta[/tex]
So, [tex]r= 2-3sin(- \theta)=2-(-3sin\theta)=2+3sin\theta=r[/tex]
Notice that the change produces the same equation.
Therefore, the given polar expression is symmetric to [tex]\theta = \frac{\pi}{2}[/tex], which is the y-axis only in the coordinate system.
So, the right answer is y-axis only.
