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Chapter 1: Problem 7

State in which quadrant or on which axis the given angle lies. $$230^{\circ}$$

Short Answer

Expert verified
Quadrant III.

Step by step solution

01

Title - Determine the Angle's Standard Position

The standard position of an angle is measured from the positive x-axis, moving counterclockwise. Measure an angle of 230掳 starting from the positive x-axis.
02

Title - Identify the Quadrants

Recall that a full circle is 360掳. The axes divide the circle into 4 quadrants: - Quadrant I: 0掳 to 90掳 - Quadrant II: 90掳 to 180掳 - Quadrant III: 180掳 to 270掳 - Quadrant IV: 270掳 to 360掳.
03

Title - Locate the Given Angle

The given angle, 230掳, falls between 180掳 and 270掳. Therefore, it lies in Quadrant III.

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Key Concepts

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

Standard Position of an Angle
To understand where an angle lies, we first need to know its standard position. An angle is said to be in the standard position when its vertex is at the origin of a coordinate plane, and one of its sides runs along the positive x-axis. Measuring an angle from this starting point, we move counterclockwise around the plane. For example, for an angle of 230掳, start measuring from the positive x-axis and move in a counterclockwise direction until you reach 230掳.
Quadrant Identification
A coordinate plane is divided into four quadrants by the x and y axes. Each quadrant covers a specific range of angles:
  • Quadrant I: From 0掳 to 90掳
  • Quadrant II: From 90掳 to 180掳
  • Quadrant III: From 180掳 to 270掳
  • Quadrant IV: From 270掳 to 360掳
When determining the location of an angle, always start measuring from the positive x-axis.
Since the given angle of 230掳 is between 180掳 and 270掳, it lies in Quadrant III. Remembering these ranges can help you quickly place any angle on the coordinate plane.
Measuring Angles in Degrees
Angles are typically measured in degrees using a protractor or similar tool. One complete revolution of a circle is 360掳. When an angle is given in degrees, you start measuring from the positive x-axis. For any angle greater than 360掳, you can subtract multiples of 360掳 until you get an equivalent angle less than 360掳.
For example:
  • If you have an angle of 370掳, subtract 360掳 (since one full rotation is 360掳) to find that 370掳 is equivalent to 10掳.
  • For a negative angle like -45掳, you add 360掳 to find the positive equivalent, giving you 315掳.
By understanding how to measure angles and convert them if necessary, you can easily place any angle in the correct quadrant.

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