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A sinusoidal function, like the sine function, is a type of periodic function which oscillates between a maximum and minimum value in a smooth, wave-like motion. The general form of a sinusoidal function is given by

• \[f(x) = a imes ext{sin}(b(x-c)) + d\]

In this equation:

• \(a\) represents the amplitude, or the height of the peaks from the central axis, determining how far the function oscillates from its mean value.
• \(b\) affects the period of the function, as it compresses or stretches the wave horizontally.
• \(c\) causes a horizontal shift, or phase shift, moving the wave left or right along the x-axis.
• \(d\) moves the wave vertically, adjusting the mean value of oscillation.

In this exercise, the sinusoidal function used is \(r = \sin(\theta / 3)\). Here, there's no amplitude shift, so the maximum and minimum values will be 1 and -1 respectively. The term \(\theta / 3\) affects the period of the wave. Understanding this component helps to anticipate the behavior of the function on the polar plane. This fundamental understanding aids in recognizing the repeating pattern that will appear in the plot as loops or petals.

The period of a sinusoidal function determines the length of one complete cycle of the wave, from start to finish. For the standard sine function \(\text{sin}(x)\),

• the period is \(2\pi\), meaning it takes \(2\pi\) radians for the function to complete one full wave cycle.

When the function is modified to \(\text{sin}(bx)\), the period gets adjusted to \(\frac{2\pi}{b}\). This results in the wave either compressing or stretching horizontally. For this exercise, the function is \(\text{sin}(\theta/3)\):

• Thus, the period becomes \(6\pi\), as \(b=\frac{1}{3}\) applies a factor which expands the cycle over the \([0, 6\pi]\) range.

This means within the interval from \(0\) to \(6\pi\), one complete cycle of the sinusoidal function occurs. It's essential to comprehend this property before plotting as it dictates how the function repeats and the nature of peaks and troughs within the specified interval. By understanding the period, we can better predict and sketch the complete curve expected in the polar coordinates.

Polar coordinates plot points on a plane using a distance from a central point (the radius \(r\)) and an angle (\(\theta\)) measured from a given direction, typically the positive x-axis. This differs from Cartesian coordinates, which rely on x and y axes to locate points.For this particular exercise, with the function \(r = \sin(\theta / 3)\), plotting in polar coordinates involves mapping how the varying radius \(r\) behaves relative to the angle \(\theta\).

• First, recognize that \(r\) changes according to the sinusoidal pattern, starting from \(0\) at \(\theta = 0\), reaching its maximum at \(\theta = 3\pi/2\) (\(r=1\)), and back to \(0\) at \(\theta = 3\pi\).
• After reaching \(0\), it continues, this time reaching a negative minimum at \(\theta = 9\pi/2\) (\(r=-1\)), before returning once more to \(0\) at \(\theta = 6\pi\).

As the angle \(\theta\) increases from \(0\) to \(6\pi\), the plot manifests as loops or petal shapes, showcasing each cycle of the sinusoidal function. These represent the extremities of \(r\), which vary positively and negatively around the polar axis.This representation is typical in polar plots, producing aesthetically pleasing and mathematically significant graphics. Understanding the pattern of a sinusoidal function in polar coordinates can provide a foundation for exploring more complex mathematical phenomena within the realm of polar graphs. Each part of the plot can often tell us much about the symmetry and periodicity of the underlying function.