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Chapter 7: Problem 17

Plot the points whose polar coordinates are given. $$\left(3,-225^{\circ}\right)$$

Short Answer

Expert verified
Convert -225掳 to 135掳 and plot the point at 3 units from the origin along the 135掳 direction.

Step by step solution

01

Understanding Polar Coordinates

Polar coordinates are given in the format \( r, \theta \), where \( r \) is the radius (distance from the origin), and \( \theta \) is the angle measured counterclockwise from the positive x-axis. The given coordinates are (3, -225掳).
02

Convert Negative Angle to Positive

Angles are usually easier to work with when they are positive. To convert -225掳 to a positive angle, keep adding 360掳 until the angle is positive: \( -225掳 + 360掳 = 135掳 \). So, -225掳 is equivalent to 135掳.
03

Plot the Point

Now that we have \( r = 3 \) and \( \theta = 135掳 \), plot the point. First, draw a line at an angle of 135掳 from the positive x-axis. Then, measure a distance of 3 units from the origin along this line, and mark the point.

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Key Concepts

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

Polar Coordinates
Polar coordinates represent a point in the plane using a radius and an angle. Instead of using x and y coordinates like in the Cartesian system, polar coordinates are written as \(r, \theta\), where \(r\) is the distance from the origin (the pole) and \( \theta\) is the angle measured from the positive x-axis. This method is particularly useful in scenarios involving circular and rotational motion. For example, given the point (3, -225掳), 3 represents the radius, and -225掳 is the angle.
Angle Conversion
Angles in polar coordinates can be measured in degrees or radians. It's important to manage angles to keep calculations straightforward.
Negative angles can be converted to positive by adding 360掳 (one full rotation). For the given point (3, -225掳), convert -225掳 to a positive angle by adding 360掳:

-225掳 + 360掳 = 135掳.

Therefore, -225掳 is equivalent to 135掳, making it easier to plot.
Plotting Points
To plot points in polar coordinates, follow these steps:
  • Draw a line from the origin at the given angle \( \theta \).
  • Measure the given radius \(r\) along that line.
  • Mark the point where the radius intersects the angle line.

For the example point (3, -225掳), immediately convert the angle -225掳 to its positive equivalent 135掳:
  • Draw a line at an angle of 135掳 from the positive x-axis.
  • Measure 3 units away from the origin along the 135掳 line.
  • Mark the point at this distance.

This method ensures an accurate representation of the point in the polar coordinate system.

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