In Exercises 27–32, select the representations that do not change the location of the given point. (7, 140°) (−7, 320°)

Verified step by step guidance

1. 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 in degrees from the positive x-axis.
2. Recall that changing the angle \(\theta\) by adding or subtracting full rotations of \(360^\circ\) does not change the point's location. So, \(\theta\) and \(\theta + 360^\circ k\) (where \(k\) is any integer) represent the same direction.
3. Note that changing the sign of \(r\) also affects the angle. Specifically, \((r, \theta)\) is equivalent to \((-r, \theta + 180^\circ)\) because moving in the opposite direction by \(180^\circ\) with a negative radius points to the same location.
4. For the point \((7, 140^\circ)\), check if representations keep the radius positive and adjust the angle by multiples of \(360^\circ\), or if the radius is negative, adjust the angle by adding \(180^\circ\) accordingly.
5. For the point \((-7, 320^\circ)\), similarly, consider adding or subtracting \(360^\circ\) to the angle, or converting the negative radius to positive by adding \(180^\circ\) to the angle, to find equivalent representations that do not change the point's location.

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance from the origin (radius) and an angle from the positive x-axis. A point is given as (r, θ), where r is the radius and θ is the angle in degrees or radians. Understanding this system is essential to interpret the given points correctly.

Equivalent Representations in Polar Coordinates

A single point in polar coordinates can have multiple representations by adjusting the radius and angle. For example, (r, θ) is equivalent to (−r, θ + 180°), or adding/subtracting full rotations (360°) to the angle. Recognizing these equivalences helps identify which transformations preserve the point's location.

Effect of Sign and Angle Changes on Point Location

Changing the sign of the radius or adding angles affects the point's position. A negative radius reverses the direction by 180°, while adding 360° to the angle results in the same direction. Understanding how these changes impact the point's location is crucial to determine which representations do not alter it.