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Understanding the absolute value of a sine function, such as in the given equation \( r = 3 \sin 2\theta \), is crucial for analyzing the behavior of trigonometric equations. The absolute value, denoted by \( | \text{ } | \), essentially measures the magnitude of a number regardless of its sign.
For the sine function, which oscillates between -1 and 1, taking the absolute value means that all the negative values become positive. Hence, the sine wave essentially mirrors onto the positive side of the y-axis for all negative values.
The result is a waveform that bobs above the x-axis between 0 and the positive amplitude of the original sine function. In the case of \( r = 3 \sin 2\theta \), since the sine function has a coefficient of 3, the maximum absolute value \( |r| \) reaches is 3, regardless of the angle \( \theta \). This occurs because the highest magnitude the sine function can achieve, before being multiplied by any coefficient, is 1. Therefore, \( \max(|r|) = |3| = 3 \).

Zeros of trigonometric functions are the values of the variable at which the function's output is precisely zero. For sine functions, such as \( \sin(\theta) \), these zeros occur at specific intervals based on the function's periodic nature.
Since the sine wave crosses the x-axis at multiples of \( \pi \), when we analyze \( r = 3 \sin 2\theta \), we are looking for the points where \( 2\theta \) would be an integer multiple of \( \pi \). That is, the function \( r \) will be zero where \( 2\theta = n\pi \), with \( n \) being any integer.
Using algebra, we can solve for \( \theta \) by dividing both sides by 2; thus, \( \theta = \frac{n\pi}{2} \). This means that for any integer value of \( n \), we obtain a zero of the function \( r \). These zeros are critical for understanding the function's turning points and when it intersects with the x-axis.

The amplitude of a sine function describes its peak vertical displacement from its central axis. Essentially, it is a measure of how 'tall' or 'short' the waves of the function are.
When dealing with the function \( r = 3 \sin 2\theta \), the amplitude can be determined by looking at the coefficient of the sine term. Here, the number 3 serves as the amplitude. In general, a sine function of the form \( A \sin(Bx) \) has an amplitude of \( |A| \).
The significance of the amplitude is that it indicates the maximum value that the function can reach above or below the central axis before reversing direction. In the context of \( r = 3 \sin 2\theta \), the amplitude tells us the sine wave will reach as high as 3 and as low as -3 on the y-axis, resulting in a full range of motion of 6 units from the peak to the trough.
This concept is vital not only in mathematics but also in various practical applications like physics, where it can represent phenomena such as wave heights in oceans or sound pressure levels in acoustics.