91Ó°ÊÓ

Polar coordinates often use radians to measure angles, but it's helpful to understand how to convert radians to degrees. This is especially useful if you are more comfortable working with degrees.
Remember that one full circle is equal to 360 degrees, which is also equal to \(2\pi\) radians. The conversion between radians and degrees can be done using the formula: \[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \]
To convert an angle from radians to degrees, you multiply by \( \frac{180}{\pi} \). For example, let's convert \( \frac{5\pi}{3} \) radians to degrees:
\[ \frac{5\pi}{3} \times \frac{180}{\pi} = 300^{\circ} \]
This tells us that \( \frac{5\pi}{3} \) radians is equivalent to 300 degrees.

Plotting a point in the polar coordinate system might seem tricky at first, but with a few steps, it becomes straightforward. Polar coordinates are in the form \( (r, \theta ) \). The first value, \( r \), represents the distance from the origin (0, 0), and the second value, \( \theta \), represents the angle measured from the positive x-axis.
To plot the point \( (5, \frac{5\pi}{3}) \):

• Start at the origin, which is the point (0, 0).
• Move 5 units away from the origin in the direction specified by the angle \( \frac{5\pi}{3} \) or 300 degrees. This angle falls in the fourth quadrant of the polar coordinate system.

By following these steps, you will have accurately plotted the polar point.

The angular coordinate in polar coordinates is denoted as \( \theta \), and it represents the angle formed with the positive x-axis. This angle can be measured in either degrees or radians.
For instance, in the polar coordinates \( \frac{5\pi}{3} \), \( \theta \) is \( \frac{5\pi}{3} \) radians, which we converted to 300 degrees.
Understanding the angular coordinate helps determine in which quadrant of the polar coordinate system the point lies:

• 0 to 90 degrees (0 to \( \frac{\pi}{2} \)) is the first quadrant.
• 90 to 180 degrees ( \( \frac{\pi}{2} \) to \( \pi \)) is the second quadrant.
• 180 to 270 degrees ( \( \pi \) to \( \frac{3\pi}{2} \)) is the third quadrant.
• 270 to 360 degrees ( \( \frac{3\pi}{2} \) to \( 2\pi \)) is the fourth quadrant.

In our example, 300 degrees places the point in the fourth quadrant.