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.