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Chapter 7: Problem 20

Plot the points whose polar coordinates are given. $$\left(-3,30^{\circ}\right)$$

Short Answer

Expert verified
Plot the point \( (3, 210^{\circ}) \) instead.

Step by step solution

01

Understanding Polar Coordinates

Polar coordinates represent a point in the plane using a radius and an angle. The form is \((r, \theta)\), where \(r\) is the distance from the origin and \( \theta \) is the angle from the positive x-axis.
02

Interpret the Given Coordinates

The given coordinates are \( (-3, 30^{\circ}) \). Here, \( r = -3 \) and \( \theta = 30^{\circ} \). The negative \( r \) value indicates the point is in the opposite direction of the angle \(30^{\circ} \).
03

Convert to Positive Radius

To plot a point with a negative radius in polar coordinates, convert \( r \) to positive by adding \( 180^{\circ} \) to the angle. So, \( 30^{\circ} + 180^{\circ} = 210^{\circ} \). Thus, the coordinates can be reinterpreted as \( (3, 210^{\circ}) \).
04

Plot the New Coordinates

On the polar grid, locate \( 210^{\circ} \). Measure out 3 units from the origin along this angle. Mark the point at this location.

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Key Concepts

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

radius and angle
In polar coordinates, every point is identified by two values: the radius and the angle. The radius, often represented as \(r\), is the distance from the origin to the point. The angle, typically denoted as \( \theta \), is measured from the positive x-axis, also known as the polar axis. It's essential to grasp how these two values work together:
  • The radius tells you how far you need to go from the origin.
  • The angle tells you the direction in which to go.
For instance, with coordinates \( (r, \theta) = (4, 45^{\circ})\), you would move 4 units from the origin at an angle of \(45^{\circ}\) from the positive x-axis.
negative radius conversion
Sometimes, polar coordinates can have a negative radius. A negative radius means you move in the exact opposite direction of the given angle. For example, if you have \((-3, 30^{\circ})\), the negative radius suggests moving 3 units in the opposite direction of \(30^{\circ}\).

To simplify plotting, convert these coordinates to a positive radius by adding \(180^{\circ}\) to the angle. This changes \(30^{\circ}\) to \(210^{\circ}\), making the coordinates \((3, 210^{\circ})\).
Now, you can plot the point easily with a positive radius.
plotting on polar grid
Plotting points on a polar grid might seem tricky at first, but it can be quite straightforward once you understand the steps. Here's how you can do it:
  • Identify the angle \(\theta\) on the polar grid. This is usually measured in degrees or radians, starting from the positive x-axis and moving counterclockwise.
  • Once you locate the right angle, measure outwards from the origin according to the radius \(r\). For example, with coordinates \( (3, 210^{\circ})\), you would find \(210^{\circ}\) and then go 3 units out from the origin along this direction.

    • It's a good idea to start with the angle first, as this sets the direction, then deal with the radius to pinpoint the exact location. Practice by plotting some points to get comfortable with switching from negative to positive radii and finding angles accurately.

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