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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 31
Chapter 5, Problem 31

In Exercises 27–32, select the representations that do not change the location of the given point. (−5, − π/4) (−5, 7π/4)

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1
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 \(2\pi\) radians (i.e., \(\theta + 2k\pi\), where \(k\) is an integer) does not change the location of the point because angles are periodic with period \(2\pi\).
Note that changing the radius \(r\) to its negative value \(-r\) and adding \(\pi\) to the angle \(\theta\) (i.e., \((-r, \theta + \pi)\)) also represents the same point, because moving in the opposite direction by \(\pi\) radians with a negative radius points to the same location.
Apply these principles to the given points: for \(( -5, -\frac{\pi}{4} )\), consider if changing the angle by \(2\pi\) or adding \(\pi\) to the angle and negating the radius results in the same point; similarly, analyze \(( -5, \frac{7\pi}{4} )\) under these transformations.
Summarize which representations keep the point's location unchanged by verifying if the transformed coordinates correspond to the same position in the plane using the above rules.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 is 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.
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Intro to Polar Coordinates

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 + π) and (r, θ + 2πk) for any integer k. Recognizing these equivalences helps identify which transformations preserve the point's location.
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Intro to Polar Coordinates

Effect of Angle and Radius Transformations on Point Location

Changing the angle by adding multiples of 2π or adjusting the radius sign with a corresponding angle shift can produce the same point. However, other transformations may move the point. Understanding how these changes affect the point's position is key to selecting representations that do not alter its location.
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Graphs of Shifted and Reflected Functions
Related Practice
Textbook Question

In Exercises 13–34, test for symmetry and then graph each polar equation. r cos θ = −3

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Textbook Question

In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 8(cos 7π/4 + i sin 7π/4)

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Textbook Question

In Exercises 30–31, find all the complex roots. Write roots in polar form with θ in degrees. The complex cube roots of 125(cos 165° + i sin 165°)

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Textbook Question

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 − 3 sin θ

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Textbook Question

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

Textbook Question

In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex fourth roots of 16 (cos 2π/3 + i sin 2π/3)