In Exercises 27–32, select the representations that do not change the location of the given point. (4, 120°) (−4, 300°)

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Understand that the point is given in polar coordinates as \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.

Recall that changing the angle \(\theta\) by adding or subtracting full rotations of \(360^\circ\) (or \(2\pi\) radians) does not change the location of the point because angles are periodic with period \(360^\circ\).

Also remember that changing the sign of \(r\) and adding \(180^\circ\) to the angle \(\theta\) results in the same point, because \((r, \theta)\) and \((-r, \theta + 180^\circ)\) represent the same location in polar coordinates.

For the point \((4, 120^\circ)\), check if the representations differ by adding or subtracting \(360^\circ\) to the angle or by changing \(r\) to \(-4\) and adding \(180^\circ\) to the angle.

For the point \((-4, 300^\circ)\), similarly check if the representations can be converted to an equivalent point by adjusting the angle by \(360^\circ\) multiples or by changing \(r\) to \(4\) and subtracting \(180^\circ\) from the angle.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar Coordinates

Polar coordinates represent points in a plane using a radius and an angle, written as (r, θ). The radius r is the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding how points are located in this system is essential for analyzing transformations.

Equivalent Polar Coordinates

A single point in polar coordinates can have multiple representations by adding or subtracting full rotations (360°) to the angle or by changing the sign of the radius and adjusting the angle by 180°. Recognizing these equivalences helps identify which representations correspond to the same point.

Effect of Transformations on Point Location

Transformations such as changing the sign of r or adding multiples of 360° to θ can alter or preserve the point's location. Understanding how these operations affect the coordinates is crucial to determine which representations keep the point unchanged.

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