A dimpled limaçon is a type of polar graph characterized by the equation $r = a + b\cos(\theta)$ or $r = a + b\sin(\theta)$, where $|a| > |b|$. It has a distinctive shape with an inner loop that does not intersect itself, creating a 'dimple'.
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The general form of the dimpled limaçon's equation is $r = a + b\cos(\theta)$ or $r = a + b\sin(\theta)$.
For it to be classified as 'dimpled,' the condition $|a| > |b|$ must hold.
If $a$ and $b$ are both positive, the graph will be symmetric about the polar axis for cosine and about the line $\theta = \frac{\pi}{2}$ for sine.
Dimpled limaçons have one 'inner loop' that creates the characteristic dimple.
The maximum radius occurs at $\theta = 0$ (or $\theta = \frac{\pi}{2}$), while the minimum radius occurs at $\theta = \pi$ (or $\theta = \frac{3\pi}{2}$).
Review Questions
What is the general form of a dimpled limaçon equation?
Under what condition does a limaçon become classified as 'dimpled'?
Describe how you can determine if a given polar equation represents a dimpled limaçon.
Related terms
Polar Coordinates: A coordinate system where each point on a plane is determined by an angle and distance from a reference point.
Cardioid: A special case of limaçons where $a = b$, producing heart-shaped curves in polar coordinates.
Inner Loop Limaçon: A type of limaçon where $|b| > |a|$, resulting in an inner loop that intersects itself.