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)

Verified step by step guidance

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.
2. 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: \(\theta_{new} = \frac{\pi}{6} + 2\pi\).
3. 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 \(\theta_{new} = \frac{\pi}{6} + \pi\).
4. 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: \(\theta_{new} = \frac{\pi}{6} - 2\pi\).
5. 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 \(\theta\).

Key Concepts

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.

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).

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.