Problem 48 - Algebra 2

Question

In which quadrant, or on which axis, does the terminal side of each angle lie? $$ -60^{\circ} $$

Step-by-Step Solution

Answer

The terminal side of the angle -60 degrees lies in the Quadrant IV.

Step 1: Understanding the problem

First, visualize a circle divided into four quadrants. Each quadrant corresponds to a set value of degrees. Quadrant I (0 to 90 degrees) Quadrant II (90 to 180 degrees) Quadrant III (180 to 270 degrees) Quadrant IV (270 degrees and above), and the cycle repeats.

Step 2: Determine rotation direction

Angles in standard position are measured from the positive x-axis, with positive angles measured counterclockwise and negative angles measured clockwise.

Step 3: Find the position of -60 degrees.

Given the angle is -60 degrees and positive rotation is counterclockwise, a negative angle implies a clockwise rotation from the positive x-axis. Now, 60-degree clockwise rotation from the positive x-axis will take the terminal side of the angle to quadrant IV.

Key Concepts

Coordinate Plane

The coordinate plane is an essential tool in geometry and trigonometry, designed to help us understand the position of angles and their corresponding points. It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis. These axes divide the plane into four distinct regions, which we call quadrants. Each quadrant is a quarter section of the plane.

• Quadrant I is where both x-coordinates and y-coordinates are positive.
• Quadrant II has negative x-coordinates but positive y-coordinates.
• Quadrant III features both x-coordinates and y-coordinates as negative.
• Quadrant IV shows positive x-coordinates but negative y-coordinates.

Understanding which quadrant an angle lies in is crucial for solving geometry problems. The terminal side of an angle helps determine its quadrant based on where it ends when the angle is drawn in standard position. In standard position, the vertex of the angle is at the origin (0,0), and one side is on the positive x-axis.

Clockwise Rotation

When discussing angles, the direction in which they are measured is important. Angles in standard position are measured from the positive x-axis. This can be done in two directions:

• Counterclockwise: This is the direction of positive angles.
• Clockwise: This is the direction of negative angles.

Clockwise rotation involves turning from the starting point on the positive x-axis towards the negative x-axis. This movement is often used to describe negative angles. For example, if you rotate -60°, you start from the positive x-axis and move in the direction of a clock’s hands, ending in a position that represents a clockwise turn from the starting point. This concept is crucial in understanding angles such as -60°, which would end in Quadrant IV after a 60° clockwise rotation.

Negative Angles

Negative angles are angles measured in the direction opposite to the way positive angles are measured, typically clockwise from the positive x-axis. Instead of adding to the angle as you rotate counterclockwise (which is standard for positive angles), you subtract as you rotate clockwise. The concept of negative angles can be seen as simply reversing the direction: when you want to rotate an angle negatively, you take a backward step on the coordinate plane. This approach is useful in various mathematical applications, such as determining the position of the terminal side in trigonometry manually. For example, an angle like -60° indicates that you begin at the positive x-axis and move 60° clockwise. Visualizing this can help determine its position, in this case ending in Quadrant IV, because you rotate towards the positive y-axis initially.