Problem 49 - Algebra 2

Question

In which quadrant, or on which axis, does the terminal side of each angle lie? $$ 0^{\circ} $$

Step-by-Step Solution

Answer

The terminal side of the angle \(0^{\circ}\) lies on the positive x-axis.

Step 1: Identifying Angle

The given angle is \(0^{\circ}\). This angle measurement implies that we have not moved from the initial side.

Step 2: Analyzing the Unit Circle

The unit circle starts at the positive x-axis (where the initial side of the angle resides) and normally moves counterclockwise. Since there's no rotation, the terminal side stays on the positive x-axis.

Step 3: Naming the Position

Since the terminal side of the angle lies along the positive x-axis, it is correct to say the angle of \(0^{\circ}\) lies on the positive x-axis.

Key Concepts

Quadrants

The coordinate plane is divided into four distinct sections known as quadrants. Each quadrant represents a specific range of angle measurements and coordinates. Here's a breakdown of what each quadrant contains:

• Quadrant I: This is where both x and y coordinates are positive. It covers angles from 0° to 90°.
• Quadrant II: In this quadrant, x coordinates are negative while y coordinates are positive. It includes angles from 90° to 180°.
• Quadrant III: Both x and y coordinates are negative here. The angles range from 180° to 270°.
• Quadrant IV: x coordinates are positive and y coordinates are negative in this section. This quadrant spans angles from 270° to 360°.

Remember, quadrants are labeled counterclockwise starting from the positive x-axis. The axes themselves do not belong to any quadrant; rather, they act as boundaries separating them.

Angles

Angles are a measure of rotation. They help us determine the direction and position of a point on the unit circle.

• Standard Position: In trigonometry, angles are usually placed in a 'standard position,' starting on the positive x-axis, known as the initial side.
• Positive and Negative Angles: Positive angles are formed by counterclockwise rotation, while negative angles are formed by clockwise rotation.
• Full Circle: A complete circle is 360°, and the angles can keep increasing or decreasing indefinitely. Angles like 0°, 360°, and 720° share the same terminal side, which can result in cycles—these angles are considered coterminal.

Understanding angles helps in visualizing and solving geometric and trigonometric problems. When given an angle, it helps to identify its position relative to the quadrants on the unit circle.

Coordinate Plane

The coordinate plane is a two-dimensional surface defined by a horizontal x-axis and a vertical y-axis intersecting at the origin. It is a foundational tool in geometry and trigonometry.

• Axes: The plane is split by the axes into four quadrants. The x-axis runs horizontally, with positive values moving right from the origin and negative values moving left. The y-axis runs vertically, with positive values moving up and negative values moving down.
• Origin: The point where the x and y axes intersect is called the origin, marked as (0,0).
• Unit Circle: A key concept often used on the coordinate plane is the unit circle. It is a circle with a radius of one unit centered at the origin. The angles and quadrants discussed previously refer back to this circle.

The coordinate plane allows us to not only visualize but also solve for various mathematical concepts, especially involving distance, midpoints, and slopes.