Problem 8 - Algebra 2 and Trigonometry

Question

In \(8-17,\) name the quadrant in which an angle of each given measure lies. $$ 25^{\circ} $$

Step-by-Step Solution

Answer

25° lies in Quadrant I.

Step 1: Understanding Quadrants

Recall that the Cartesian plane is divided into four quadrants. When moving counterclockwise from the positive x-axis: - Quadrant I: 0° to 90° - Quadrant II: 90° to 180° - Quadrant III: 180° to 270° - Quadrant IV: 270° to 360°.

Step 2: Determine the Quadrant for 25°

Since 25° lies between 0° and 90°, it falls into Quadrant I.

Key Concepts

Quadrants

The concept of quadrants is crucial when working with trigonometric angles on the Cartesian plane. We divide the plane into four distinct regions, or quadrants, to help us pinpoint the position of an angle in relation to the axes.

• Quadrant I: This is where the positive x-axis begins and extends to the positive y-axis, covering angles from 0° to 90°.
• Quadrant II: This quadrant lies between the positive y-axis and the negative x-axis, covering angles from 90° to 180°.
• Quadrant III: This extends from the negative x-axis to the negative y-axis, covering angles from 180° to 270°.
• Quadrant IV: In this quadrant, we move from the negative y-axis back to the positive x-axis, completing the circle with angles from 270° to 360°.

Moving counterclockwise from the positive x-axis, each of these quadrants helps define where an angle's terminal side will land, making it easier to identify and work with angles on the plane.

Cartesian Plane

The Cartesian plane is a two-dimensional plane created by two perpendicular number lines intersecting at a point called the origin. This system is named after the French mathematician René Descartes. The two intersecting axes divide the plane into the four quadrants we previously discussed.
The horizontal number line is called the x-axis, and the vertical one is the y-axis. The point where these axes meet, with coordinates (0,0), is the origin. From here, every point on the plane can be described using an ordered pair \(x, y\).

• The x-coordinates are measured along the horizontal axis: to the right for positive values and to the left for negative values.
• The y-coordinates are measured along the vertical axis: upward for positive values and downward for negative values.

The Cartesian plane is a powerful tool for understanding mathematical concepts, providing a clear and structured way to visualize complex relationships between numbers, angles, and coordinates.

Degree Measurement

Angles are an essential part of trigonometry and geometry, and degree measurement is a fundamental way of expressing the size of an angle. A complete circle is divided into 360 degrees, allowing us to measure angles and locate them accurately on the Cartesian plane.
Each degree can be subdivided into 60 smaller units called minutes, which can further be divided into 60 seconds. While this level of precision isn’t always necessary, it can be useful for very specific applications in fields like navigation and astronomy.

• Positive and Negative Angles: Positive angles are measured counterclockwise from the positive x-axis, whereas negative angles are measured clockwise.
• Finding Angle Position: By knowing the degree measurement, such as 25°, we can determine the exact quadrant in which the angle lies using our knowledge of the four quadrants.

Understanding degree measurements not only helps us in identifying quadrants but also plays a crucial role in trigonometric calculations and solving geometric problems.