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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
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Tutti i libri di testoBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 21
Capitolo 5, Problema 21

In Exercises 21–26, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. Then find another representation of this point in which
a. r>0, 2π < θ < 4π.
b. r<0, 0. < θ < 2π.
c. r>0, −2π. < θ < 0.


(5, π/6)

Guida verificata passo dopo passo
1
Step 1: Understand the given point in polar coordinates: \(r = 5\), \(\theta = \frac{\pi}{6}\). This means the point is 5 units from the origin at an angle of \(\frac{\pi}{6}\) radians measured counterclockwise from the positive x-axis.
Step 2: For part (a), find another representation where \(r > 0\) and \(2\pi < \theta < 4\pi\). Since the angle can be coterminal by adding multiples of \(2\pi\), add \(2\pi\) to the original angle: \(\theta_{new} = \frac{\pi}{6} + 2\pi\).
Step 3: For part (b), find a representation where \(r < 0\) and \(0 < \theta < 2\pi\). To get a negative radius, use the fact that \((r, \theta)\) is equivalent to \((-r, \theta + \pi)\). So, set \(r_{new} = -5\) and \(\theta_{new} = \frac{\pi}{6} + \pi\).
Step 4: For part (c), find a representation where \(r > 0\) and \(-2\pi < \theta < 0\). To get a negative angle coterminal with \(\frac{\pi}{6}\), subtract \(2\pi\) from the original angle: \(\theta_{new} = \frac{\pi}{6} - 2\pi\).
Step 5: Summarize the new representations: (a) \((5, \frac{\pi}{6} + 2\pi)\), (b) \((-5, \frac{\pi}{6} + \pi)\), and (c) \((5, \frac{\pi}{6} - 2\pi)\). These satisfy the given conditions for \(r\) and \(\theta\).

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Polar Coordinates and Their Representation

Polar coordinates represent points in a plane using a radius (r) and an angle (θ) measured from the positive x-axis. Each point can have multiple representations by adjusting r and θ, reflecting the same location in different ways. Understanding how to plot and interpret these coordinates is fundamental to solving the problem.
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Intro to Polar Coordinates

Angle Measurement and Periodicity

Angles in polar coordinates are periodic with a period of 2π, meaning adding or subtracting multiples of 2π results in the same direction. This property allows for multiple angle representations of the same point, especially when adjusting θ to fit within specified intervals like (2π, 4π) or (−2π, 0).
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Period of Sine and Cosine Functions

Negative Radius and Angle Adjustments

A negative radius (r < 0) in polar coordinates points in the opposite direction of the angle θ, effectively shifting the point by π radians. This concept is crucial when finding alternate representations with negative r, as it requires adjusting θ accordingly to maintain the point's position.
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Coterminal Angles
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