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

Verified step by step guidance

1. Step 1: Understand the problem. We are given two points in polar coordinates: (4, 120°) and (−4, 300°). We need to determine which representations do not change the location of these points.
2. Step 2: Recall that in polar coordinates, a point (r, θ) can be represented as (r, θ + 360°k) or (-r, θ + 180° + 360°k), where k is an integer.
3. Step 3: For the point (4, 120°), consider the equivalent representations: (4, 120° + 360°k) and (-4, 120° + 180° + 360°k).
4. Step 4: For the point (−4, 300°), consider the equivalent representations: (-4, 300° + 360°k) and (4, 300° + 180° + 360°k).
5. Step 5: Identify which of these representations match the given points without changing their location.

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). The format is (r, θ), where 'r' is the radial distance and 'θ' is the angle in degrees or radians. Understanding polar coordinates is essential for analyzing how points can be represented in different ways without changing their location.

Angle Measurement

Angles in polar coordinates can be measured in degrees or radians, and they can be expressed in multiple equivalent forms. For example, an angle of 120° is equivalent to 120° + 360°n for any integer n. Recognizing these equivalences is crucial for determining whether different representations of a point in polar coordinates refer to the same location.

Symmetry in Polar Coordinates

Polar coordinates exhibit symmetry, particularly with respect to the origin and the axes. For instance, a point (r, θ) has a corresponding point (-r, θ + 180°), which reflects it across the origin. Understanding this symmetry helps in identifying representations that maintain the same location in the polar coordinate system.