Problem 68 - Precalculus

Question

State in which quadrant or on which axis each angle with the given measure in standard position would lie. $$11.4$$

Step-by-Step Solution

Answer

The angle 11.4 degrees lies in the first quadrant.

Step 1: Locate the Reference Angle

The given angle is 11.4 degrees. Angles are measured counter-clockwise from the positive x-axis. An angle of 11.4 degrees originates close to the positive x-axis.

Step 2: Determine the Quadrant or Axis

Since 11.4 degrees is between 0 degrees and 90 degrees, it lies in the first quadrant, where angles range from 0 to 90 degrees.

Key Concepts

Angle Measurement

In trigonometry, angle measurement is a fundamental concept that helps us understand how angles are defined and located within a coordinate system. Typically, angles are measured in degrees. Mathematicians often start measuring from the positive x-axis and move counterclockwise.

Angles measured in degrees can range from 0 to 360, describing a full circle. Here are some key points to remember:

• Positive angles move counterclockwise.
• Negative angles move clockwise.

For example, an angle of 11.4 degrees means it opens slightly off the positive x-axis in a counterclockwise direction. Being familiar with how an angle presents itself on a coordinate plane is essential to analyzing its properties and determining in which part of the plane it will appear.

Standard Position

The notion of standard position is something you'll encounter frequently when dealing with angles in mathematics, especially in trigonometry. An angle is in standard position if its vertex is located at the origin of a coordinate plane and its initial side (starting position) aligns along the positive x-axis.

Here's what you should keep in mind:

• The origin acts as the pivot point for the angle.
• The initial side helps in determining the direction and size of the angle.

By understanding standard position, you are better equipped to calculate and identify the region of the coordinate plane where an angle will fall. For instance, an angle of 11.4 degrees in standard position starts on the positive x-axis and sweeps into the first quadrant. Recognizing the standard position is key to effectively locate angles.

Quadrants in Coordinate Plane

The coordinate plane is divided into four sections known as quadrants. Each quadrant helps in identifying where a particular angle lies, based on its measurement.

The coordinate plane's quadrants are:

• First Quadrant: Covers angles from 0 to 90 degrees.
• Second Quadrant: Encompasses angles from 90 to 180 degrees.
• Third Quadrant: Includes angles from 180 to 270 degrees.
• Fourth Quadrant: Encompasses angles from 270 to 360 degrees.

To find an angle's quadrant, you compare its measure to these ranges. For example, an angle of 11.4 degrees is situated in the first quadrant. Understanding the quadrants is essential for solving problems related to angles in trigonometry, as certain properties and values such as sine or cosine can vary between quadrants.