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Chapter 5: Problem 25

Name the quadrant in which each angle lies. $$300^{\circ}$$

Short Answer

Expert verified
Quadrant IV

Step by step solution

01

Convert the angle to its standard position

An angle greater than 360掳 can be converted to its equivalent angle within the first 360掳 by subtracting 360. However, since 300掳 is already less than 360掳, it is already in its standard position.
02

Identify the quadrants on the coordinate plane

The coordinate plane is divided into four quadrants: Quadrant I (0掳 to 90掳), Quadrant II (90掳 to 180掳), Quadrant III (180掳 to 270掳), and Quadrant IV (270掳 to 360掳).
03

Determine the quadrant for 300掳

Since 300掳 lies between 270掳 and 360掳, it falls in Quadrant IV.

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Key Concepts

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

Angle Conversion
When working with angles, it's often necessary to convert them to fit within the standard interval of 0掳 to 360掳. This process is called angle conversion.

If an angle is more than 360掳, you can always subtract 360掳 from it until it fits within this interval. For example, an angle of 370掳 can be converted by subtracting 360掳, giving you 10掳.

It's important to do this because most angle-related concepts and problems assume that angles are within this first circle of 360掳. However, in our example with 300掳, it is already less than 360掳, so no conversion is needed.
Coordinate Plane Quadrants
The coordinate plane is a two-dimensional plane divided by the x-axis and y-axis into four regions called quadrants. These quadrants help us identify the location of points and angles more easily.

Here's how each quadrant is defined:
  • Quadrant I (0掳 to 90掳): Points in this quadrant have both positive x and y values.
  • Quadrant II (90掳 to 180掳): Points in this quadrant have a negative x value and a positive y value.
  • Quadrant III (180掳 to 270掳): Points in this quadrant have both negative x and y values.
  • Quadrant IV (270掳 to 360掳): Points in this quadrant have a positive x value and a negative y value.
In our example, since 300掳 is between 270掳 and 360掳, it falls in Quadrant IV.
Standard Position of Angles
Angles are often discussed in their 'standard position' on the coordinate plane. This is a specific way to visualize and define angles to make calculations easier.

Here's what you need to know about standard position:
  • An angle is in standard position if its vertex is at the origin (0,0) of the coordinate plane.
  • The initial side of the angle lies along the positive x-axis.
  • Angles are measured from this initial side, with positive angles measured counterclockwise and negative angles measured clockwise.
Since 300掳 is less than 360掳 and measured from the positive x-axis, it is already in its standard position. This allows us to easily determine it falls in Quadrant IV.

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