How do you graph #r=3+2sintheta#?

The Socratic graph, graph{x^2+y^2-3 sqrt(x^2+y^2)-2y = 0}, is an immediate graph.

This limacon can be rotated clockwise around the pole r = 0 by means of

This limacon's Cartesian form is

To graph ( r = 3 + 2\sin(\theta) ), where ( r ) is the distance from the origin and ( \theta ) is the angle measured from the positive x-axis in standard position, you can follow these steps:

1. Plot points:

   • Choose values of ( \theta ) (usually ranging from ( 0 ) to ( 2\pi )) and calculate the corresponding values of ( r ).
   • Substitute the values of ( \theta ) into the equation ( r = 3 + 2\sin(\theta) ) to find the corresponding values of ( r ).
   • Plot these points on a polar coordinate system.

2. Connect the points:

   • Once you have plotted several points, connect them smoothly to form the graph of the equation.

3. Consider symmetry:

   • Since ( \sin(\theta) ) is an odd function, the graph will exhibit symmetry about the origin.
   • Check for any other symmetries based on the equation, such as symmetry about the line ( \theta = \pi ).

4. Label key points and features:

   • Label key points such as the intercepts with the axes, maximum and minimum points, and any other notable features.

5. Add any necessary details:

   • Include arrows indicating the direction of increasing ( \theta ) if needed.
   • Label the axes and indicate the scale if the graph is to be used for precise measurements.

By following these steps, you can graph the equation ( r = 3 + 2\sin(\theta) ) on a polar coordinate system accurately.