Problem 45 - Algebra 2

Question

In which quadrant, or on which axis, does the terminal side of each angle lie? $$ 150^{\circ} $$

Step-by-Step Solution

Answer

The terminal side of the angle 150 degrees lies in Quadrant 2.

Step 1: Identify the quadrant ranges

Begin by remembering that a circle has 360 degrees and the quadrants are divided as follows: Quadrant 1 (0-90°), Quadrant 2 (90-180°), Quadrant 3 (180-270°) and Quadrant 4 (270-360°). Also the positive direction is counterclockwise. Beneath the horizontal axis values are negative.

Step 2: Comparing the given angle to quadrant ranges

Next, look at the given angle, which is 150°. Compare this to the quadrant ranges identified in step 1. The angle 150° falls within the range for Quadrant 2, which is from 90° to 180°.

Step 3: Drawing the conclusion

Considering that the angle 150° lies between 90° and 180°, the conclusion is that the terminal side of the 150° angle lies in Quadrant 2.

Key Concepts

Coordinate Plane

The coordinate plane is a fundamental concept in geometry and trigonometry. It is made up of two number lines: one horizontal (x-axis) and one vertical (y-axis). These two lines intersect at the origin, which has the coordinates (0, 0). The coordinate plane is used to locate points in a two-dimensional space via these perpendicular lines that divide the plane into four quadrants.

• Quadrant I: Positive x and y coordinates.
• Quadrant II: Negative x and positive y coordinates.
• Quadrant III: Negative x and y coordinates.
• Quadrant IV: Positive x and negative y coordinates.

Together, these quadrants provide a full set of coordinates to describe any point or vector including angles in standard position on the coordinate plane.

Terminal Side

When an angle is drawn in standard position on the coordinate plane, its vertex is located at the origin and one of its sides lies along the positive x-axis. The other side, which rotates to form the angle, is referred to as the terminal side.
The terminal side can rotate counterclockwise for positive angles and clockwise for negative angles. This is essential in determining where an angle lies on the coordinate plane. For example, a 150° angle has its terminal side reaching into Quadrant II. Understanding the terminal side helps visualize the angle's position in relation to the coordinate axes.

Degree Measurement

Degree measurement is a system for expressing the size of an angle. Angles are commonly measured in degrees, where a full rotation around a circle is 360 degrees. This system divides the circle into 360 equal parts.
Degrees are convenient for everyday usage and are often preferred in navigation, maps, and geometry. For calculating where an angle falls on the coordinate plane, the degrees help identify the angle's position with respect to each quadrant. For instance, the 150° angle is part of Quadrant II, as it fits within the 90° to 180° range that characterizes this quadrant.

Quadrant Identification

Identifying the quadrant in which an angle resides involves comparing the angle's degree measurement to the ranges of the four quadrants on the coordinate plane. Each quadrant spans 90°:

• Quadrant I: 0 to 90°
• Quadrant II: 90 to 180°
• Quadrant III: 180 to 270°
• Quadrant IV: 270 to 360°

By determining where an angle like 150° falls within these intervals, you can easily conclude that it lies in Quadrant II since 150° is greater than 90° and less than 180°. This method is a straightforward way to understand the rotational position of an angle on the coordinate plane.