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A limaçon is a type of polar curve. It can be identified by its defining equation: \( r = a \, (1 + b \, ext{function} \, \theta) \). The equation may include either a sine or cosine term. This influences the curve's orientation relative to the pole, which is the origin in polar coordinates.
³¢¾±³¾²¹Ã§´Ç²Ôs come in various forms, including those with loops, ones that do not touch the pole, or limaçons that appear like a cardioid. These forms depend on the ratio of \( a \) to \( b \):

• If \( |b| < a \), the limaçon has an inner loop.
• If \( |b| = a \), the limaçon forms a cardioid shape.
• If \( |b| > a \), the curve is an ordinary limaçon without a loop.

In this exercise, the equation \( r = 3(1 + \, \sin \theta) \) is in the form of a limaçon with a loop, as \( a \) is greater than the coefficient of \( \sin \theta \) (which is usually "1"). Recognizing this allows you to anticipate the loop in the graph.

The polar equation is a way to represent curves in a two-dimensional plane using polar coordinates. Instead of using the typical \( x \) and \( y \) coordinates, polar equations use \( r \) (the radial distance from the origin) and \( \theta \) (the angle from the positive x-axis). This is beneficial when dealing with circular and spiral patterns.For understanding polar equations, it's important to note:

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

The given polar equation, \( r = 3(1 + \, \sin \theta) \), immediately indicates a dependency on \( \theta \) shaped by the sine function. The nature of \( \sin \theta \) creates a varying value for \( r \), leading to more complex curves like limaçons.

Curve sketching in polar coordinates starts by understanding how \( r \) changes with \( \theta \). To sketch the curve from \( r = 3(1 + \sin \theta) \), we begin by finding crucial points. This involves substituting notable angles into the equation such as \( \theta = 0, \frac{\pi}{2}, \pi, \) and \( \frac{3\pi}{2} \). At these angles, \( r \) is calculated to determine the points:

• At \( \theta = 0 \), \( r = 3 \).
• At \( \theta = \frac{\pi}{2} \), \( r = 6 \).
• At \( \theta = \pi \), \( r = 3 \).
• At \( \theta = \frac{3\pi}{2} \), \( r = 0 \).

Once these are calculated, the points can be plotted on the polar plane. The curve should smoothly connect these points. To ensure accuracy:

• Identify symmetry in the curve, as it might reflect over an axis.
• Recognize the loop caused by polar function properties.
• Seek to reflect smooth, continuous connections through the polar plane.

This limaçon requires careful placement of points, especially noting where the inner loop forms, providing a complete and accurate sketch.