Problem 125 - 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. $$ \left(3, \frac{\pi}{6}\right) $$

Step-by-Step Solution

Answer

The point is 3 units from the origin along a 30-degree angle from the positive x-axis.

Step 1: Understand Polar Coordinates

Polar coordinates are given as \((r, \theta)\), where \(r\) is the radial distance from the origin, and \(\theta\) is the angular component in radians measured from the positive x-axis. Here, \(r = 3\) and \(\theta = \frac{\pi}{6}\).

Step 2: Construct the Angle \(\theta\)

Convert the angle \(\theta\) from radians into degrees if preferred for easier construction. \(\theta = \frac{\pi}{6}\) radians is equivalent to 30 degrees. Draw a ray from the origin forming a 30-degree angle with the positive x-axis.

Step 3: Mark Off the Distance \(r\)

Using the ray constructed at 30 degrees, mark a distance of \(3\) units from the origin along this ray. Ensure the measurement is accurate to reflect \(r = 3\).

Step 4: Plot the Point

The point with polar coordinates \((3, \frac{\pi}{6})\) is located 3 units away from the origin along the 30-degree ray. Draw a point at this location on the polar grid.

Key Concepts

Radial Distance

In polar coordinates, the radial distance is a crucial component that describes how far a point is from the origin, also known as the pole. This distance is analogous to the radius of a circle formed around the origin. In the provided exercise, the point is represented by the polar coordinates \( (3, \frac{\pi}{6}) \), where \( r = 3 \) is the radial distance. This means that the point is exactly 3 units away from the origin.

• The radial distance is always a non-negative value. It essentially sets the size of the circle on which the point lies.
• In the context of the exercise, after determining the angular component, the radial distance helps us locate the point accurately by telling us how far along the constructed ray the point should be marked.
• Visualizing the radial distance as the length of a line from the origin to the point can aid in understanding polar coordinates better.

By accurately marking this distance on the polar grid, we ensure the point's position is correctly plotted.

Angular Component

The angular component in polar coordinates is represented by \( \theta \), which describes the rotation needed from the positive x-axis to reach the direction of the point. It is measured in radians, though converting to degrees is often more intuitive. In the problem, \( \theta = \frac{\pi}{6} \) tells us that this direction creates an angle of 30 degrees with the positive x-axis.

• The angular component allows for determining the direction of the ray from the origin, which is essential before applying the radial distance.
• Angles in polar coordinates can be thought of in terms of full rotations, where \( 2\pi \) radians equal 360 degrees. This perspective aids in converting between radians and degrees efficiently.
• Constructing this angle correctly ensures that the ray, and subsequently the point, is placed in the exact direction required.

Understanding and manipulating the angular component accurately is vital in plotting polar coordinates effectively.

Polar Grid

A polar grid is a special kind of graph where points are plotted based on their radial distance and angular component. It resembles a spider web or target, where concentric circles represent equal radial distances, and lines radiating from the center represent different angles.

• The polar grid serves as the canvas upon which polar coordinates can be visualized and understood better.
• Each concentric circle corresponds to a specific distance from the center, helping to gauge the radial component of a point.
• Ray lines, typically marked by different angles, aid in accurately plotting the directional component of the point.

Using a polar grid is essential for graphing points given in polar coordinates, as it provides a clear and structured way to understand the relationship between radial distances and angles. It makes it easier to see how these two components work together to pinpoint a location in this coordinate system.