Problem 127 - Calculus Volume 3

Question

Plot the point whose polar coordinates are given by first constructing the angle \(\theta\) and then marking off the distance \(r\) along the ray. $$ \left(0, \frac{7 \pi}{6}\right) $$

Step-by-Step Solution

Answer

The point (0, \(\frac{7\pi}{6}\)) is at the origin.

Step 1: Understanding Polar Coordinates

In polar coordinates, each point is defined by a radius \(r\) and an angle \(\theta\). The point \((r, \theta) = (0, \frac{7\pi}{6})\) has a radius \(r = 0\) and an angle \(\theta = \frac{7\pi}{6}\).

Step 2: Identifying the Radius

Since the radius is \(r = 0\), the point is at the origin, regardless of the angle \(\theta\). The plot involves marking this point at the origin on the graph.

Step 3: Visualizing the Angle

The angle \(\theta = \frac{7\pi}{6}\) corresponds to \(210^\circ\) in standard position (measured counterclockwise from the positive x-axis). However, since \(r = 0\), this angle won't affect the position of the point.

Step 4: Conclusion about the Location

Given that \(r = 0\), any angle \(\theta\) will result in locating the point at the origin. Therefore, plot the point at the origin \((0, 0)\).

Key Concepts

Radius in Polar Coordinates

In polar coordinates, the radius often denoted by \( r \), describes the distance from the origin to a point. It tells you how far you need to "travel" to reach your desired point in the polar coordinate system.

- If \( r = 0 \), the point lies exactly at the origin, regardless of the angle \( \theta \). No travel needed!- If \( r > 0 \), you move \( r \) units away from the origin in the direction specified by the angle \( \theta \).

The radius is crucial because it combines with the angle to uniquely define a location in polar coordinates. Keep in mind that increasing the radius moves the point further away from the center, but does not affect the direction toward which you move.

Angle in Polar Coordinates

The angle \( \theta \) in polar coordinates dictates the direction in which you "travel" from the origin. It's measured in radians or degrees, moving counterclockwise from the positive x-axis. For example, an angle of \( \theta = \frac{7\pi}{6} \) radians equals 210 degrees.

Guidelines for determining the angle include:

• Angles move counterclockwise from the starting zero angle (positive x-axis).
• An angle larger than \( 2\pi \) represents more than one full revolution and brings you back to a previously encountered direction.
• An angle can be negative, indicating a clockwise direction.

Despite \( \theta \), if \( r = 0 \), the direction does not affect the location of the point, which remains at the origin.

Graphing Polar Coordinates

Graphing polar coordinates involves plotting points based on their radius and angle. Here's how to do it:

1. **Start from the origin**: Begin your graph at the center or 'origin', where the radial lines intersect.2. **Determine the angle \( \theta \)**: Measure \( \theta \) counterclockwise from the positive x-axis. This direction sets the "line" along which the point will be placed, though it may not move if \( r = 0 \).3. **Plot the radius \( r \)**: From the origin, move outwards along the angle line by a distance equivalent to the radius \( r \). If \( r = 0 \), stay put at the origin without moving.
Graphing polar coordinates can offer a circular and often intuitive view of plotting, which is especially useful in fields such as electromagnetism, physics, and engineering. The combination of radius and angle is powerful, providing a comprehensive method to locate points on the plane.