Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

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Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problema 28

Capitolo 5, Problema 28

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

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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.

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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.

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Percorso guidato

05:32

Intro to Polar Coordinates

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.

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Percorso guidato

05:32

Intro to Polar Coordinates

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.

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Percorso guidato

5:25

Introduction to Transformations

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