Problem 46 - Basic Technical Mathematics with Calculus

Question

Designate each angle by the quadrant in which the terminal side lies, or as a quadrantal angle. $$180^{\circ}, 92^{\circ}$$

Step-by-Step Solution

Answer

180° is a quadrantal angle; 92° lies in the second quadrant.

Step 1: Understand Quadrantal Angles

A quadrantal angle is an angle whose terminal side lies on one of the axes (x or y). For example, angles like \(0^{\circ}, 90^{\circ}, 180^{\circ},\) and \(270^{\circ}\) are quadrantal angles.

Step 2: Identify the Quadrant for 180°

Since \(180^{\circ}\) is one of the common angles that lies exactly on the axes, it does not belong to any quadrant. It lies on the negative x-axis.

Step 3: Identify the Quadrant for 92°

To find out which quadrant the angle \(92^{\circ}\) belongs to, determine in which range of degrees (0 to 90, 90 to 180, 180 to 270, 270 to 360) it falls. \(92^{\circ}\) is greater than \(90^{\circ}\) and less than \(180^{\circ}\), so the angle's terminal side lies in the second quadrant.

Key Concepts

Quadrantal Angles

In trigonometry, quadrantal angles hold a special place. These angles have terminal sides that rest directly on the x-axis or y-axis of the coordinate system.

• Examples of quadrantal angles include \(0^{\circ}, 90^{\circ}, 180^{\circ},\) and \(270^{\circ}\).
• Because their terminal sides are aligned with the axes, they do not fall into any of the four quadrants.
• Instead, they separate these quadrants and serve as critical points for measuring rotational directions and periodic functions in trigonometry.

Understanding these angles is crucial, as they mark the boundaries of the four quadrants. For example, when considering the angle \(180^{\circ}\), it is often referred to as lying on the negative x-axis.
These distinct angles are vital in defining the behavior of trigonometric functions, such as sine and cosine, which possess unique values at these points.

Coordinate System

The coordinate system is a fundamental concept in mathematics, acting as the stage for angles and their measurements.
Largely depicted as a grid with horizontal (x-axis) and vertical (y-axis) lines, this system is used to locate points and describe angles.

• A typical coordinate grid divides into four quadrants, numbered counterclockwise from the positive x-axis. This numbering helps in determining the position of angle terminal sides.
• Quadrant I is where both x and y are positive, Quadrant II where x is negative and y is positive, Quadrant III where both are negative, and Quadrant IV where x is positive and y is negative.

The x and y axes are the foundations of this system, and any angle's terminal side can be dissected in relation to these axes. Understanding how quadrants are divided and where specific angles fall within these quadrants is essential for solving trigonometry problems.

Angle Measurement

Angle measurement is an integral part of trigonometry that enables us to determine the position and behavior of an angle's terminal side.

• Angles are typically measured in degrees, a complete circle amounting to \(360^{\circ}\).
• The measurement starts from the positive x-axis and moves counterclockwise.

To ascertain where an angle lies, you must relate it to the four quadrants or recognize it as a quadrantal angle. For example, an angle like \(92^{\circ}\) is greater than \(90^{\circ}\) but less than \(180^{\circ}\).
Thus, it lies in Quadrant II.Furthermore, knowing these distinctions assists in predicting the behavior of trigonometric ratios at specific angles—vital for both theoretical analyses and practical applications in physics, engineering, and real-world problem-solving scenarios.