Problem 28 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Plot the point \(3, \frac{5 \pi}{3}\).

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 angle measured in radians from the positive x-axis.

Step 2: Identify the Given Coordinates

The given polar coordinates are \(-3, \frac{2 \pi}{3}\). In this case, \r = -3\ and \theta = \frac{2 \pi}{3}\.

Step 3: Interpret the Negative Radius

A negative radius means that the point lies in the direction opposite to \theta\. Therefore, we plot the point \(3, \frac{5 \pi}{3}\). This angle, \frac{5 \pi}{3}\, is equivalent because \frac{5 \pi}{3} = \frac{6 \pi}{3} - \frac{ \pi}{3} = 2 \pi - \frac{ \pi}{3}\.

Step 4: Plot the Point

To plot \(3, \frac{5 \pi}{3}\), first measure the angle \frac{5 \pi}{3}\ which is in the fourth quadrant. Then, move 3 units from the origin in the direction of the angle.

Key Concepts

radial distance

In polar coordinates, the radial distance is an essential concept. It is the distance from the origin, typically denoted as \( r \). This distance helps us determine how far a point is from the center of the polar coordinate system.
Unlike Cartesian coordinates which use \(x\) and \(y\) values, polar coordinates use a single value for distance and a separate value for direction.
Here are the key points about radial distance:

angle in radians

The angle in radians, denoted as \( \theta \), specifies the direction of the point from the origin. It is measured from the positive x-axis (the right direction).
Radians are a unit of angle measurement, where \( 2\pi \) radians is equivalent to 360 degrees. This means \( \pi \) radians is 180 degrees, and so on.
For example, an angle of \( \frac{2\pi}{3} \) radians means moving counterclockwise from the positive x-axis by two-thirds of 180 degrees (which is 120 degrees). Here are additional points:

negative radius interpretation

A negative radius can be slightly tricky. It indicates that the point is in the opposite direction to the given angle \( \theta \). Instead of plotting at distance \( r \) at angle \( \theta \), you plot at distance \( |r| \) at an angle of \( \theta + \pi \) radians.
In the exercise, we had:\[ r = -3, \theta = \frac{2\pi}{3} \]
This translates to an equivalent point at a distance of 3 units and angle \( \frac{5\pi}{3} \) radians, because:
\[ \frac{5\pi}{3} = 2\pi - \frac{\pi}{3} \]
This adjustment moves the direction to the opposite side of the angle.

plotting points in polar coordinates

Sometimes students find plotting in polar coordinates challenging. To plot a point given in polar coordinates \( r, \theta \):

• First, measure the angle \( \theta \).
• Second, move \( r \) units from the origin along the direction of the angle.

For our example, plotting \(3, \frac{5\pi}{3}\) involves:
1. Measuring the angle \( \frac{5\pi}{3} \) radians in the fourth quadrant.
2. Moving a distance of 3 units along that direction.
Visualizing it on polar grid makes it easier to understand.