4. (8 points) Sketch $r=1+\sin \theta$ in rectangular coordinates for $0 \leq \theta \leq 2 \pi$. Then use the rectangular graph as a table of values to sketch $r=1+\sin \theta$ in polar coordinates. Be sure to use arrows and numbers to show the direction of the curve and how each portion of the rectangular curve corresponds to the polar curve.

Step 1 ：First, we need to sketch the graph of \(r=1+\sin \theta\) in rectangular coordinates for \(0 \leq \theta \leq 2 \pi\).

Step 2 ：The rectangular coordinates of a point on this graph is given by \(x = r \cos \theta\) and \(y = r \sin \theta\).

Step 3 ：Substitute \(r = 1 + \sin \theta\) into the equations, we get \(x = (1 + \sin \theta) \cos \theta\) and \(y = (1 + \sin \theta) \sin \theta\).

Step 4 ：Then, we can use these equations to sketch the graph in rectangular coordinates.

Step 5 ：Next, we need to sketch the graph of \(r=1+\sin \theta\) in polar coordinates.

Step 6 ：We can use the rectangular graph as a table of values to sketch the polar graph.

Step 7 ：Be sure to use arrows and numbers to show the direction of the curve and how each portion of the rectangular curve corresponds to the polar curve.

Step 8 ：Finally, check whether your results meet the requirements of the problem.