sketch the graph of $r = 3+3sin\theta$.

Explanation:

Step1: Identify the type of polar curve

The equation $r = a + b\sin\theta$ with $a = b=3$ represents a cardioid.

Step2: Find key - points

When $\theta = 0$, $r=3 + 3\sin(0)=3$. When $\theta=\frac{\pi}{2}$, $r=3 + 3\sin(\frac{\pi}{2})=6$. When $\theta=\pi$, $r=3+3\sin(\pi)=3$. When $\theta=\frac{3\pi}{2}$, $r=3 + 3\sin(\frac{3\pi}{2})=0$.

Step3: Sketch the curve

Plot the points $(3,0)$, $(6,\frac{\pi}{2})$, $(3,\pi)$, $(0,\frac{3\pi}{2})$ and then draw a smooth heart - shaped curve (cardioid) passing through these points. The cardioid is symmetric about the line $\theta=\frac{\pi}{2}$.

Answer:

Sketch a cardioid passing through the points $(3,0)$, $(6,\frac{\pi}{2})$, $(3,\pi)$, $(0,\frac{3\pi}{2})$ and symmetric about the line $\theta=\frac{\pi}{2}$.