Problem 12 - Algebra 2 and Trigonometry

Question

In \(8-17,\) name the quadrant in which an angle of each given measure lies. $$ -75^{\circ} $$

Step-by-Step Solution

Answer

The angle \(-75^{\circ}\) lies in Quadrant IV.

Step 1: Understanding Angle Signs and Directions

Angles are measured from the positive x-axis, with positive angles rotating counter-clockwise and negative angles rotating clockwise. A negative angle like \(-75^{\circ}\) indicates a clockwise rotation starting from the positive x-axis.

Step 2: Determining Quadrants

The Cartesian plane is divided into four quadrants: Quadrant I (0 to 90°), Quadrant II (90 to 180°), Quadrant III (180 to 270°), and Quadrant IV (270 to 360° or -90° to 0°). Since we rotate clockwise \(75^{\circ}\) from the positive x-axis, \(-75^{\circ}\) falls just before reaching the negative y-axis (or 0°), which places it in Quadrant IV.

Key Concepts

Quadrant Determination

Determining which quadrant an angle falls into involves understanding both the degrees of the angle and the direction in which it is measured. The Cartesian plane is divided into four quadrants:

• Quadrant I: Angles between 0° and 90° (counter-clockwise)
• Quadrant II: Angles between 90° and 180°
• Quadrant III: Angles between 180° and 270°
• Quadrant IV: Angles between 270° and 360° or, equivalently, between -90° and 0° (clockwise)

When you are given an angle, you start measuring from the positive x-axis. Positive angles move counter-clockwise, while negative angles move clockwise. This helps us pinpoint exactly which quadrant the angle lies in on the Cartesian plane with just a quick calculation or observation.

Negative Angles

Negative angles are simply a way of measuring rotation in a clockwise direction. When dealing with angles, the direction is key:

• Positive angles are those measured counter-clockwise from the positive x-axis.
• Negative angles rotate clockwise from the same starting point.

For example, an angle of \(-75^{\circ}\) means you rotate 75 degrees in the clockwise direction from the positive x-axis. This results in a different positioning on the Cartesian plane compared to its positive counterpart. Recognizing how to handle and interpret negative angles is essential for accurate quadrant determination and other angle-related problems.

Cartesian Plane

The Cartesian plane is a two-dimensional plane formed by two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin (0,0), dividing the plane into four distinct areas known as quadrants:

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, y is negative.

Each quadrant represents a space where certain mathematical rules about signs and rotation apply. Being familiar with the Cartesian plane helps when determining angles and their corresponding quadrants, as it provides a foundational framework for understanding rotation and directionality in mathematics.