Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Plot the points whose polar coordinates follow. For each point, give four other pairs of polar coordinates, two with positive \(r\) and two with negative \(r\). (a) \(\left(1, \frac{1}{2} \pi\right)\) (b) \(\left(-1, \frac{1}{4} \pi\right)\) (c) \(\left(\sqrt{2},-\frac{1}{3} \pi\right)\) (d) \(\left(-\sqrt{2}, \frac{5}{2} \pi\right)\)

Step-by-Step Solution

Answer

Convert polar coordinates by adjusting angles and flipping signs of radius while keeping trigonometric properties.

Step 1: Understanding Polar Coordinates

Polar coordinates are in the form of \((r, \theta)\), where \(r\) represents the radius or distance from the origin, and \(\theta\) represents the angle measured from the positive x-axis. These coordinates can be adjusted by adding/subtracting multiples of \(2\pi\) to \(\theta\). Also, a negative radius \(r\) flips the location to the opposite side of the origin.

Step 2: Handling Point (a)

The given point is \((1, \frac{1}{2}\pi)\). This means it is 1 unit away from the origin at an angle of \(\frac{1}{2}\pi\).1. Two equivalent points with positive \(r\): \((1, -\frac{3}{2}\pi)\) and \((1, \frac{5}{2}\pi)\).2. Two equivalent points with negative \(r\): \((-1, -\frac{1}{2}\pi)\) and \((-1, \frac{3}{2}\pi)\).

Step 3: Handling Point (b)

The given point is \((-1, \frac{1}{4}\pi)\). Originally, at \((1, \frac{1}{4}\pi)\), flipping across the origin with -1, it maps to \((-1, \frac{1}{4}\pi)\).1. Two equivalent points with positive \(r\): \((1, \frac{5}{4}\pi)\) and \((1, -\frac{7}{4}\pi)\).2. Two equivalent points with negative \(r\): \((-1, \frac{9}{4}\pi)\) and \((-1, -\frac{7}{4}\pi)\).

Step 4: Handling Point (c)

The given point is \((\sqrt{2}, -\frac{1}{3}\pi)\).1. Two equivalent points with positive \(r\): \((\sqrt{2}, \frac{5}{3}\pi)\) and \((\sqrt{2}, -\frac{1}{3}\pi + 2\pi) = (\sqrt{2}, \frac{5}{3}\pi)\).2. Two equivalent points with negative \(r\): \((-\sqrt{2}, \frac{2}{3}\pi)\) and \((-\sqrt{2}, -\frac{4}{3}\pi)\).

Step 5: Handling Point (d)

The given point is \((-\sqrt{2}, \frac{5}{2}\pi)\).1. Two equivalent points with positive \(r\): \((\sqrt{2}, -\frac{1}{2}\pi)\) and \((\sqrt{2}, \frac{3}{2}\pi)\).2. Two equivalent points with negative \(r\): \((-\sqrt{2}, \frac{1}{2}\pi)\) and \((-\sqrt{2}, -\frac{5}{2}\pi + 2\pi) = (-\sqrt{2}, -\frac{1}{2}\pi)\).

Key Concepts

Polar Coordinate System

The polar coordinate system is a fascinating way to represent the location of points on a plane. Instead of using the traditional Cartesian (x, y) system, this method uses two parameters: a radius and an angle. The radius, denoted as \(r\), indicates how far the point is from a central origin point, similar to the center of a circle. The angle, \(\theta\), shows the direction in which to move from the center along the circumference.
In polar coordinates, each point is identified by a pair \((r, \theta)\). While the radius \(r\) is always considered in terms of distance, it can be negative, which reflects the point across the origin. The angle \(\theta\) is measured from the positive x-axis, and angles are typically expressed in radians.
The polar coordinate system is extremely useful in various fields, such as engineering and physics, especially when dealing with circular motions or phenomena.

Converting Polar Coordinates

Converting polar coordinates involves transforming polar descriptions into Cartesian descriptions and vice versa. This can be essential for situations where one form might be more convenient than the other.
To convert a polar point \((r, \theta)\) to Cartesian coordinates \((x, y)\), use the formulas:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

This conversion leverages the definitions of sine and cosine to project the radius \(r\) onto the x and y axes.
Conversely, to transform from Cartesian \((x, y)\) to polar coordinates, you calculate:

• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \tan^{-1}(\frac{y}{x})\)

Understanding these conversions can open up easier solutions to specific geometric problems.

Equivalent Polar Coordinates

In the polar coordinate system, a single point can have multiple equivalent representations. This is because you can modify the angle \(\theta\) by adding or subtracting multiples of \(2\pi\), without altering the point's location.
For example, the point \((r, \theta)\) can also be written as \((r, \theta + 2\pi n)\) where \(n\) is an integer. This flexibility is because angles in polar do not reset after a full rotation, unlike angles in a fixed 0 to \(360^{\circ}\) system.

• For positive \(r\), equivalent points can be found directly by altering \(\theta\).
• For negative \(r\), the point reflects across the origin, necessitating an additional adjustment to \(\theta\) of \(\pi\) radians for equivalent representation.

This ability to generate equivalent coordinates is useful in many mathematical computations and graphical representations.

Plotting Polar Coordinates

Plotting polar coordinates involves placing points on a polar grid, which looks like a series of concentric circles (for varying \(r\)) and radial lines (for varying \(\theta\)).
To graph a point \((r, \theta)\):

• Begin at the origin, then move \(\theta\) radians from the positive x-axis. This dictates the direction.
• Proceed \(r\) units along this direction to reach the point.

With practice, plotting becomes intuitive, allowing for quick visual representation of data, like navigation paths or waveforms. Remember, the beauty of the polar plot is its ability to simplify complex circular patterns into a single, coherent image. A negative \(r\) value essentially means extending \(-r\) units in the direction opposite to \(\theta\). This can create symmetries that are sometimes more visually appealing than Cartesian plots.