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Polar coordinates are a way of expressing points in a plane using a pair of numbers. Unlike Cartesian coordinates, which use horizontal and vertical positions (\(x\) and \(y\) axes), polar coordinates express a point in terms of:

• \(r\): The radial distance from the origin or pole.
• \(\theta\): The angle from a reference direction, typically the positive \(x\)-axis.

This system is highly beneficial for dealing with curves and figures that exhibit radial or circular symmetry, such as circles and spirals.

When studying curves in polar coordinates, each point is described by how far it is from the center and the angle it makes. For example, the equation \(r = 1 + 3 \cos \theta\) in our exercise describes a limaçon, a popular type of polar curve that can have special features like an inner loop.

The cosine function, \(\cos \theta\), is a fundamental trigonometric function that relates an angle in a right triangle to the ratio of the adjacent side over the hypotenuse. In the context of polar equations, the cosine function helps describe how the radius \(r\) changes with respect to the angle \(\theta\).

The cosine function is periodic, with a period of \(2\pi\). This means it repeats its values every \(2\pi\) radians. A cosine value ranges from \(-1\) to \(1\), and the function is symmetric around the vertical axis, so it affects the shape and features of polar graphs significantly.

For our equation, \(1 + 3 \cos \theta\), the cosine term governs how much \(r\) varies as \(\theta\) increases, leading to the formation of an inner loop when conditions are met such as \(|a| < |b|\).

Trigonometric equations involve trigonometric functions of angles and are used to find specific angle values that satisfy the equation.

The typical format of such an equation is \(\cos \theta = c\). The solution to these equations involves finding all angles \(\theta\) within given bounds that satisfy this condition.

• When solving \(\cos \theta = -\frac{1}{3}\), we employ the inverse cosine function to obtain potential solutions.
• These solutions are given as infinite series, \(\theta = \pm \cos^{-1}(c) + 2k\pi\), where \(k\) is any integer.

By considering only the solutions in the interval \([0, 2\pi]\), we narrow it down to those relevant for our polar graph.

Polar equations use polar coordinates to describe curves in a plane. The equation \(r = a + b \cos \theta\) defines a specific curve known as a limaçon. In this form,

• \(a\) and \(b\) are constants that determine the size and specific traits of the limaçon.
• The expression \(b \cos \theta\) describes how \(r\) varies with \(\theta\), ultimately affecting the curve's shape.

The limaçon shows an inner loop if \(|a| < |b|\), as we see in our problem where \(a = 1\) and \(b = 3\). This results in interesting geometrical features exploring how the distance from the origin, \(r\), diminishes to zero within a specific \(\theta\) interval.

Understanding angle intervals is crucial when working with trigonometric equations. They help determine when specific features in a polar graph occur.

For the limaçon \(r = 1 + 3 \cos \theta\), we calculated \(\cos \theta = -\frac{1}{3}\). To find specific \(\theta\)-intervals for the inner loop, we look for angles one period at a time, typically \([0, 2\pi]\).

• The principal angle is \(\theta = \cos^{-1}(-\frac{1}{3})\).
• Given the symmetry of cosine, another solution in the same range is \(2\pi - \cos^{-1}(-\frac{1}{3})\).

The resulting interval where the inner loop is visible in the polar graph is approximately \([1.9106, 4.3726]\) radians. Thus, angle intervals tell us precisely where the curve displays unique behavior like looping.