In Exercises 27–32, select the representations that do not change the location of the given point. (−6, 3π) (6, −π)

Verified step by step guidance

1. Understand that the problem involves identifying which representations of points in polar coordinates correspond to the same location. Polar coordinates are given as \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.
2. Recall that a point in polar coordinates can have multiple representations that correspond to the same location. For example, adding or subtracting \(2\pi\) to the angle \(\theta\) does not change the point's location because angles differing by full rotations point in the same direction.
3. Also remember that changing the sign of \(r\) and adding \(\pi\) to the angle \(\theta\) gives an equivalent point: \((r, \theta) = (-r, \theta + \pi)\). This means that \((r, \theta)\) and \((-r, \theta + \pi)\) represent the same point.
4. For the point \((-6, 3\pi)\), consider converting it using the rule above: change \(r\) to positive \(6\) and add \(\pi\) to the angle \(3\pi\). This will give an equivalent representation with positive radius.
5. For the point \((6, -\pi)\), consider adding \(2\pi\) to the angle \(-\pi\) to get an equivalent angle in the standard range \([0, 2\pi)\). This will help identify if the location remains the same.

Key Concepts

Polar Coordinates and Their Representation

Polar coordinates represent points in the plane using a radius and an angle, written as (r, θ). The radius r indicates the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding how points are plotted in polar form is essential to analyze transformations that affect their location.

Equivalent Polar Coordinates

A single point in polar coordinates can have multiple representations due to periodicity of angles and sign changes in radius. For example, (r, θ) is equivalent to (r, θ + 2πk) for any integer k, and also to (−r, θ + π + 2πk). Recognizing these equivalences helps identify which transformations preserve the point's location.

Effect of Changing Sign of Radius and Angle

Changing the sign of the radius or adding/subtracting multiples of π or 2π to the angle can produce different coordinate pairs representing the same point. For instance, (−r, θ) corresponds to (r, θ + π), so understanding these relationships is key to determining which representations do not change the point's position.