91Ó°ÊÓ

In polar coordinates, the radial distance, denoted as \( r \), represents how far away a point is from the origin or the center of the polar coordinate system. It is always measured as a straight line from the origin to the point in question.
This concept is akin to measuring the radius of a circle.
Think of it as the "length" part of the coordinate that stretches outward from the center.
Here's how to understand radial distance better:

• A positive radial distance means the point is in the same direction as the angle \( \theta \).
• A negative radial distance flips the direction, placing the point opposite to where \( \theta \) points.

Radial distance helps determine the size and scale of plots on the polar graph. When plotting points like \( \left( \frac{1}{3}, \frac{3\pi}{2} \right) \), the distance of \( \frac{1}{3} \) indicates that the point is just a third of a unit away from the center, indicating a rather small circle placement.

The angle in polar coordinates is given in radians, which is a measure of the arc along the circle as it extends from the positive x-axis. In our exercise, the angle \( \theta = \frac{3\pi}{2} \) corresponds to 270 degrees, making the point fall directly on the negative y-axis.
Understanding angles in radians is crucial as it dictates the direction from the origin:

• Radian is a unit of angular measure used in many areas of mathematics.
• One full rotation around a circle is \( 2\pi \) radians, equivalent to 360 degrees.
• The value of \( \theta \) starts from the positive x-axis and moves counterclockwise.
• Angles greater than \( 2\pi \) continue beyond a full circle, while negative angles rotate clockwise.

To visually interpret an angle like \( \frac{3\pi}{2} \), picture moving three-quarters of the way around the unit circle, which takes you straight downward.

Coordinate transformation in polar coordinates refers to changing the way a point is expressed by altering either its radial distance or angle, often to fit specific conditions. This is essential for expressing a point in multiple forms without changing its actual location.
Common transformations include reflections and shifts of the angle:

• To transform a point with a negative radial distance, add \( \pi \) to the original angle \( \theta \).
• For an angle falling outside \( 0 \) to \( 2\pi \), add or subtract \( 2\pi \) to bring it back within this range.

For example, by transforming \( \left( \frac{1}{3}, \frac{3\pi}{2} \right) \) so the distance is negative but the angle stays between 0 and \( 2\pi \), you get \( \left( -\frac{1}{3}, \frac{\pi}{2} \right) \), effectively reflecting it to the opposite side.

Converting from polar to Cartesian coordinates is a crucial skill, linking the polar system with the more widely used rectangular coordinate system. This conversion allows for plotting points using x and y coordinates, familiar to many students.
The conversion formulas are simple yet powerful:

• To find the x-coordinate, use \( x = r \cdot \cos(\theta) \).
• For the y-coordinate, use \( y = r \cdot \sin(\theta) \).

Using the original point \( \left( \frac{1}{3}, \frac{3\pi}{2} \right) \), conversion to Cartesian coordinates involves:

1. Calculating \( x = \frac{1}{3} \cdot \cos\left(\frac{3\pi}{2}\right) = 0 \) because the cosine of 270 degrees is zero.
2. Calculating \( y = \frac{1}{3} \cdot \sin\left(\frac{3\pi}{2}\right) = -\frac{1}{3} \), since the sine of 270 degrees is -1.

Thus, the Cartesian coordinates of the point are \( (0, -\frac{1}{3}) \), mapping it to a more intuitive location on a familiar x-y grid.