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A limaçon is a type of graph in polar coordinates. It resembles a distorted circle and is defined by an equation of the form \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). The terms \(a\) and \(b\) in the equation determine its shape.
For our given problem \( r = \sqrt{3} + \cos \theta \), \( a \) is \( \sqrt{3} \) and \( b \) is 1.
This configuration results in a limaçon without an inner loop. Limaçons can have various appearances:

• Inner Loop: Occurs when \(|b| > |a|\).
• Cardioid: Forms when \(|a| = |b|\).
• Dimpled Limaçon: Happens when \(|a| > |b|\), but the shape is still slightly indented.
• Convex Limaçon: Appears as a rounded shape when \(|a| >> |b|\).

In our exercise, \( |a| = \sqrt{3} \) is greater than \( |b| = 1 \).
Hence, the limaçon is free of inner loops and takes on a more convex, or outwardly curved, form.

Polar coordinates provide a different way to describe locations of points in a plane. Unlike the Cartesian system that uses \( (x, y) \) coordinates, polar coordinates rely on a distance \( r \) from the origin and an angle \( \theta \) from the positive x-axis.

• \( r \) represents how far the point is from the origin (the pole).
• \( \theta \) indicates the direction of \( r \) and is measured in radians.

When plotting a graph like a limaçon in polar coordinates, you first calculate the radius \( r \) for different values of \( \theta \).
You mark these points on the graph using the calculated \( r \) and its corresponding \( \theta \).
These distinctive points help sketch the curve accurately.
The given polar equation \( r = \sqrt{3} + \cos \theta \) requires evaluating \( r \) at specific angles such as \( \theta = 0, \frac{\pi}{2}, \pi, \text{and} \frac{3\pi}{2} \).
This approach ensures the complete shape of the limaçon is well-represented on the graph.

Sketching a polar graph, especially a limaçon, involves several key steps to ensure accuracy. First, we need to identify the key points through substitution of angles into the polar equation to find their corresponding \( r \) values.
This data allows us to pinpoint specific locations on the polar plane.

• Key Points: These are determined by plugging in angles like \( \theta = 0 \) or \( \pi \), which help shape the graph's structure.
• Symmetry: Due to the cosine term in our equation, the graph has symmetry about the x-axis.
• Shape: Limaçons are characteristically rounded and their size and shape fluctuate with different values of \( \theta \).

To sketch the limaçon, plot each key point with its \( r \) distance correctly relative to \( \theta \).
Connect these points smoothly to capture the limaçon's unique curvature.
Remember, in a limaçon that uses \( \cos \theta \), the widest part of the shape appears directed towards the right side or positive \( x \)-axis. This effect is due to the positive direction of \( \cos \theta \) affecting the values of \( r \).
Carefully drafting these details ensures a graph that reflects the true characteristics of the specified polar equation.