Inner-loop limaçons are a type of polar graph represented by the equation $r = a + b \cos(\theta)$ or $r = a + b \sin(\theta)$ with $|a| < |b|$. These graphs feature a distinct inner loop due to the absolute value of $a$ being less than that of $b$.
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Inner-loop limaçons have a characteristic inner loop when $|a| < |b|$.
$r = a + b \cos(\theta)$ and $r = a + b \sin(\theta)$ are the standard forms for these curves.
The length of the inner loop is determined by the difference between $|b|$ and $|a|$.
The graph intersects the pole (origin) twice, creating the loop.
For inner-loop limaçons, if $b > 0$, the loop is in the direction of $\cos(\theta)$ or $\sin(\theta)$. If $b < 0$, it is in the opposite direction.
Review Questions
What are the conditions on coefficients 'a' and 'b' for an inner-loop limaçon to form?
Write down the standard polar equations for an inner-loop limaçon.
How does changing the values of 'a' and 'b' affect the size and position of an inner-loop limaçon?
Related terms
Polar Coordinates: A coordinate system where each point on a plane is determined by its distance from a reference point and its angle from a reference direction.
Limaçon: A family of curves described by polar equations of the form $r = a + b \cos(\theta)$ or $r = a + b \sin(\theta)$. The shape varies based on values of 'a' and 'b'.
Cardioid: $r = a + a \cos(\theta)$ or $r = a + a \sin(\theta)$, which forms a heart-shaped curve without an inner loop.