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Chapter 7: Problem 16

Plot the points whose polar coordinates are given. $$(1,-2 \pi / 3)$$

Short Answer

Expert verified
Plot at 1 unit from the origin in the 240掳 (clockwise \frac{2\pi}{3} radians) direction.

Step by step solution

01

Understand Polar Coordinates

Polar coordinates are represented as \(r, \theta\), where \(r\) is the radius and \(\theta\) is the angle in radians. To plot these, we need to identify these two components from the given coordinates.
02

Identify Components of the Given Coordinates

The given polar coordinates are \(1, -\frac{2\pi}{3}\). Here, \(r = 1\) and \(\theta = -\frac{2\pi}{3}\).
03

Understand the Angle

Angles in polar coordinates are measured counterclockwise from the positive x-axis. A negative angle indicates a clockwise direction. \(\theta = -\frac{2\pi}{3}\) means rotating \(\frac{2\pi}{3}\) radians clockwise.
04

Locate the Angle

Clockwise rotation of \(\frac{2\pi}{3}\) radians equates to \(240掳\).
05

Plot the Radius

From the origin (0,0), move 1 unit in the direction of \(240掳\). Since the radius \(r = 1\), it is exactly 1 unit away from the origin.
06

Mark the Point

After moving 1 unit along the \(240掳\) direction, place a point. This marks the location of \(1, -\frac{2\pi}{3}\) in the polar coordinate system.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radius
Radius is a key concept in polar coordinates. In polar representation, the radius (denoted as \(r\)) measures the distance from the origin (or center) of the coordinate system to the point in question.
For the given coordinates \((1, -\frac{2\pi}{3})\), the radius \(r\) is \(1\). This means that the point is exactly 1 unit away from the origin.
Imagine you're standing at the center of a circle (the origin) and you take a step in any direction. That step is your radius, and it defines a circle around you. Longer steps (larger \(r\)) mean a bigger circle.
Here, our radius being 1 tells us that the point lies on the circumference of a circle with radius 1, centered at the origin.
Angle in Radians
Angles in polar coordinates define the direction in which you move from the origin. The angle (denoted as \(\theta\)) is measured in radians, a standard unit of angular measure.
In this exercise, the angle \(\theta = -\frac{2\pi}{3}\) means you should move clockwise around the circle. Negative angles indicate a clockwise direction, whereas positive angles indicate counterclockwise movement.
To understand \(\theta = -\frac{2\pi}{3}\), convert it to degrees if needed: \(-\frac{2\pi}{3}\) radians is equal to \(-120掳\), or 120 degrees clockwise from the positive x-axis.
Plotting Points
Plotting points in polar coordinates is like connecting the dots in a specific direction and distance.
First, locate your origin (0,0). Then, using your radius of 1 unit, position yourself 1 unit away.
Next, factor in your angle. For \(\theta = -\frac{2\pi}{ 3}\) or 240 degrees, picture the standard x and y axes. Start from the positive x-axis, then move clockwise by 120 degrees.
Finally, mark the point where this direction intersects your radial distance. This completes your plotting of the point using polar coordinates.

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