In the study of polar coordinates, it's essential to understand that a single point can be represented by multiple ordered pairs. This concept is similar to coterminal angles in the unit circle, where angles like \( \frac{\pi}{6} \) and \( \frac{13\pi}{6} \) point to the same location. In polar coordinates, a point is defined by an ordered pair \( (r, \theta) \), where \( r \) is the radius and \( \theta \) is the angle. To find different representations of the same point, we can apply the same principles used for coterminal angles.
For any point represented as \( (r, \theta) \), we can generate additional ordered pairs by adding or subtracting multiples of \( 2\pi \) to the angle \( \theta \). This can be expressed mathematically as:
\( (r, \theta) \) is coterminal with \( (r, \theta + 2\pi n) \) for any integer \( n \).
Additionally, if we want to change the sign of \( r \) (making it negative), we must adjust the angle \( \theta \) by adding \( \pi \) to it. This ensures that the point remains in the same location. The relationship can be summarized as:
\( (r, \theta) \) is equivalent to \( (-r, \theta + \pi) \).
For example, starting with the point \( (4, \frac{\pi}{6}) \), we can find another representation by keeping \( r \) the same and adjusting \( \theta \). If we subtract \( 2\pi \) from \( \frac{\pi}{6} \), we get:
\( (4, \frac{\pi}{6} - 2\pi) = (4, -\frac{11\pi}{6}) \).
To find a representation with a negative \( r \), we set \( r = -4 \) and keep \( \theta \) within the interval \( [0, 2\pi] \). The original angle \( \frac{\pi}{6} \) is already in this range, but we need to adjust it for the negative radius:
\( (-4, \frac{\pi}{6} + \pi) = (-4, \frac{7\pi}{6}) \).
When working with polar coordinates, it's crucial to remember these transformations to accurately find multiple ordered pairs that correspond to the same point. This understanding not only enhances your grasp of polar coordinates but also strengthens your overall mathematical skills in dealing with angles and their representations.
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