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Chapter 2: Problem 13

State in which quadrant or on which axis each of the following angles with given measure in standard position would lie. $$ 12.34^{\circ} $$

Short Answer

Expert verified
The angle 12.34掳 lies in Quadrant I.

Step by step solution

01

Understanding the Coordinate Plane

The coordinate plane consists of four quadrants. Quadrant I ranges from 0掳 to 90掳, Quadrant II from 90掳 to 180掳, Quadrant III from 180掳 to 270掳, and Quadrant IV from 270掳 to 360掳. An angle is measured counterclockwise from the positive x-axis.
02

Identify the Angle Position

The given angle is 12.34掳. Determine in which range it falls. Since 0掳 < 12.34掳 < 90掳, 12.34掳 is within the range of Quadrant I.
03

Conclusion Based on Quadrant Analysis

Since the angle 12.34掳 lies between 0掳 and 90掳, it is in Quadrant I. Angles in this range point towards the top right portion of the coordinate plane.

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Key Concepts

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

Coordinate Plane
The coordinate plane is a two-dimensional surface where we can plot points representing numbers. It is made up of two axes:
  • The horizontal axis is called the x-axis.
  • The vertical axis is called the y-axis.
Together, these axes divide the plane into four sections known as quadrants.
These quadrants are numbered in a counterclockwise direction starting from the upper right:
  • Quadrant I: top right
  • Quadrant II: top left
  • Quadrant III: bottom left
  • Quadrant IV: bottom right
Each quadrant contains points with specific positive or negative values for x and y coordinates.
Understanding the coordinate plane helps us determine where angles in standard position lie.
Standard Position
In trigonometry, an angle is said to be in standard position when its vertex is at the origin of the coordinate plane. The initial side of the angle is on the positive x-axis.
To measure the angle, you rotate the terminal side starting from the initial side. Angles are measured by this rotation, either going in a counterclockwise or clockwise direction:
  • A counterclockwise rotation results in a positive angle.
  • A clockwise rotation results in a negative angle.
This standardization helps in easily identifying which quadrant an angle belongs to based on its measurement.
For example, an angle like 12.34掳 starts from the positive x-axis and extends into Quadrant I.
Angle Measurement
Angle measurement is fundamental in locating the position of an angle in trigonometry. Usually measured in degrees, a full circle is 360掳.
Here are some key regions to remember:
  • 0掳 - 90掳 lies in Quadrant I.
  • 90掳 - 180掳 lies in Quadrant II.
  • 180掳 - 270掳 lies in Quadrant III.
  • 270掳 - 360掳 lies in Quadrant IV.
Angles start at the positive x-axis, known as the initial side.
From here, moving counterclockwise determines where the angle opens. For instance, an angle of 12.34掳 is found within Quadrant I, meaning it is relatively small and points upward and to the right of the coordinate plane.

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