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Chapter 5: Problem 23

Name the quadrant in which each angle lies. $$-125^{\circ}$$

Short Answer

Expert verified
-125 degrees lies in the third quadrant.

Step by step solution

01

Understand the Angle Notation

Angles measured in the negative direction are measured clockwise from the positive x-axis.
02

Locate the Angle in a Full Rotation

First, we need to determine how far \( -125^{\circ} \) is from 0. Since each full rotation is 360 degrees, subtract the given angle from 360: \(360^{\circ} - 125^{\circ} = 235^{\circ} \).
03

Identify the Quadrant

An angle of \( 235^{\circ} \) lies between \( 180^{\circ} \) and \( 270^{\circ} \), which means it is in the third quadrant. Therefore, \( -125^{\circ} \) also lies in the third quadrant.

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Key Concepts

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

angle measurement
When we talk about angle measurement, it's essential to understand the basics first. Angles are typically measured in degrees (掳). A full circle is divided into 360 degrees.
To measure an angle, we start from the positive x-axis (to the right) and move counterclockwise for positive angles. For example, a 90掳 angle points straight up. An angle of 180掳 points to the left.
Measuring angles helps us determine their direction and position within the circular coordinate plane. This is important when we move on to concepts like negative angles or specific quadrants.
negative angles
Negative angles are just a way of measuring angles in a different direction. Instead of moving counterclockwise, we measure clockwise from the positive x-axis.
For instance, a -45掳 angle means we start from the positive x-axis and move 45 degrees clockwise. This can sometimes be confusing, but the concept is simple once you get the hang of it.
Think of negative angles as just another way to describe the same positions as positive angles. For example, -45掳 and 315掳 describe the same spot on the coordinate plane because -45掳 is 45掳 clockwise and 315掳 is 315掳 counterclockwise, both ending up in the same place.
full rotation
A full rotation in angle measurement refers to moving a complete 360 degrees around the circle.
This is important to understand where an angle sits within the coordinate plane. If we start at 0掳 and go a complete circle, we end back at 0掳 or 360掳.
So, if we have an angle like 720掳, that means we've gone around twice because 720掳 is 360掳 multiplied by 2. Similarly, for negative angles, a full rotation backwards (clockwise) would be -360掳. Knowing this helps us find where angles land, especially when they are beyond the first 360掳 range.
third quadrant
The coordinate plane is divided into four quadrants.
- The first quadrant is where both x and y are positive.
- The second quadrant is where x is negative, and y is positive.
- The third quadrant is where both x and y are negative.
- The fourth quadrant is where x is positive, and y is negative.

In this exercise, when we found the angle -125掳, we converted it to 235掳. This falls between 180掳 and 270掳.
Angles in this range are in the third quadrant. Hence, -125掳 lies in the third quadrant. By understanding this concept, it becomes easier to pinpoint where any angle is located on the coordinate plane.

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