Indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (3, −135°)

검증된 단계별 안내

Recall that polar coordinates are given in the form \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle measured counterclockwise from the positive x-axis.

Note that the given point is \((3, -135^\circ)\). Since the angle is negative, convert it to a positive angle by adding \(360^\circ\): \(-135^\circ + 360^\circ = 225^\circ\).

Plot the point by moving a distance of 3 units from the origin in the direction of \(225^\circ\). This angle lies in the third quadrant of the Cartesian plane.

Identify which labeled point (A, B, C, or D) on the graph corresponds to the location at \(3\) units from the origin along the \(225^\circ\) direction.

Confirm your choice by checking the position of the point relative to the axes and the given labels on the graph.

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주요 개념

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Polar Coordinates System

Polar coordinates represent points in a plane using a distance from the origin (radius) and an angle measured from the positive x-axis. Each point is given as (r, θ), where r is the radius and θ is the angle in degrees or radians.

Negative Angles in Polar Coordinates

A negative angle in polar coordinates means measuring the angle clockwise from the positive x-axis. For example, −135° is equivalent to rotating 135° clockwise, which can be converted to a positive angle by adding 360°, resulting in 225°.

Plotting Points with Negative Radius

If the radius r is positive, the point lies in the direction of the angle θ. If r is negative, the point is plotted in the opposite direction of θ by rotating 180°. In this question, r is positive, so the point lies 3 units from the origin at the angle −135°.