Which polar equation corresponds to a limaçon?

The polar equation that corresponds to a limaçon is characterized by its specific form, which typically includes terms that can be expressed as ( r = a + b \cos(\theta) ) or ( r = a + b \sin(\theta) ), where ( a ) and ( b ) are non-negative constants.

In the case of the equation ( r = 3 + 3 \cos(\theta) ), we can identify this as a limaçon because it matches the standard form with ( a = 3 ) and ( b = 3 ). The presence of the cosine function indicates that the limaçon will have a symmetry along the polar axis, and the parameters ( a ) and ( b ) determine the shape and whether it has an inner loop, is dimpled, or is convex.

When analyzing the specific characteristics of this equation, if ( a ) is equal to ( b ), as it is here, the limaçon will specifically have an inner loop. This scenario exemplifies how polar coordinates can create complex shapes based on simple linear relationships between ( r ) and ( \theta ).

The other equations listed do not showcase the same properties: one represents a rose