Problem 29 - Algebra and Trigonometry

Question

Plot each point given in polar coordinates. $$ \left(4,-\frac{2 \pi}{3}\right) $$

Step-by-Step Solution

Answer

Plot 4 units from the origin at \(\frac{4\pi}{3}\) radians in the third quadrant.

Step 1: Understand Polar Coordinates

Polar coordinates are represented as \(r, \theta\), where \(r\) is the radius and \(\theta\) is the angle in radians. Our given coordinates are \(4, -\frac{2 \pi}{3}\).

Step 2: Identify the Radius

The radius \(r\) is the distance from the origin to the point. In this case, \(r = 4\). This means our point is 4 units away from the origin.

Step 3: Identify the Angle

The angle \(\theta\) is \(-\frac{2\pi}{3}\). This is a negative angle, which means it is measured clockwise from the positive x-axis.

Step 4: Convert to Positive Angle

To find the equivalent positive angle, we add \(2\pi\) to \(-\frac{2\pi}{3}\). \[\theta = -\frac{2\pi}{3} + 2\pi = \frac{4\pi}{3}\] The positive angle \(\frac{4\pi}{3}\) is in the third quadrant.

Step 5: Plot the Point

Now plot the point \(4, \frac{4\pi}{3}\) by moving 4 units from the origin along the direction of the angle \(\frac{4\pi}{3}\). This places the point in the third quadrant.

Key Concepts

radius

In polar coordinates, the radius (often denoted as \(r\)) is a key concept. It represents the distance from the origin (the center point) to the plotted point. For example, in the coordinates \((4, -\frac{2 \pi}{3})\), the radius is 4. This means that the point is exactly 4 units away from the origin, regardless of the direction in which it is placed. Think of the radius as the length of a straight line drawn from the center to the point. It helps determine how far you need to go from the origin to reach the specific point on the polar plane.

angle in radians

When working with polar coordinates, the angle \(\theta\) is just as important as the radius. The angle tells you the direction in which you need to move from the origin to plot the point. Angles in polar coordinates are usually measured in radians, a unit based on the radius of a circle. For example, when given \(-\frac{2 \pi}{3}\), this angle is measured clockwise from the positive x-axis because it is negative. To better understand the location of angles in radians, visualize a circle divided into parts: one full revolution (circle) is \(2\pi\) radians.
For our example problem, \(-\frac{2 \pi}{3}\) can be converted to a positive angle by adding \(2\pi\), making it \(\frac{4\pi}{3}\), which places it in the third quadrant.

plotting points in polar coordinates

Plotting points in polar coordinates involves using both the radius and the angle. Here's how you can plot the point \((4, \frac{4\pi}{3})\):

• Start at the origin (0,0).
• Measure the angle of \(\frac{4\pi}{3}\) counterclockwise from the positive x-axis. Since \(\frac{4\pi}{3}\) is in the third quadrant, this angle visually helps to locate the direction.
• Move 4 units away from the origin in the direction of the angle you measured.

This method ensures you find the precise spot in the polar coordinate system. It's a systematic way to locate any given point using its radius and angle.

coordinate transformation

Coordinate transformation is the process of converting coordinates from one system to another. In the case of polar coordinates, it sometimes involves converting negative angles to positive ones to better understand their location on the polar plane. For instance, in the given example, we started with the polar coordinates \((4, -\frac{2\pi}{3})\). Due to the negative angle, calculating its positive equivalent helps in easier plotting:

• Negative angle \(-\frac{2\pi}{3}\) measures clockwise.
• To find the positive equivalent, add \(2\pi\): \(-\frac{2\pi}{3} + 2\pi = \frac{4\pi}{3}\).

This transformation places the point accurately in its respective quadrant, ensuring correct placement in the polar coordinate system.