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

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.