Problem 45 - Basic Technical Mathematics with Calculus

Question

The given angles are in standard position. Designate each angle by the quadrant in which the terminal side lies, or as a quadrantal angle. $$31^{\circ}, 310^{\circ}$$

Step-by-Step Solution

Answer

31° is in Quadrant I; 310° is in Quadrant IV.

Step 1: Understand the angle positions

Angles are generally categorized based on their terminal side location after standard position rotation. The standard position is when the angle's vertex is at the origin and initial side on the positive x-axis. Quadrantal angles are directly on one of the axes: 0°, 90°, 180°, 270°, or 360°. Other angles fall into one of four quadrants. Quadrants are I (0° to 90°), II (90° to 180°), III (180° to 270°), and IV (270° to 360°).

Step 2: Classify the 31° angle

The angle 31° is measured counterclockwise from the positive x-axis. Since 31° is between 0° and 90°, it is in Quadrant I, where all values are positive. The terminal side of the angle lies within this quadrant.

Step 3: Classify the 310° angle

Angle 310° is measured counterclockwise from the x-axis. It's more than 270° and less than 360°, which places it in Quadrant IV. This quadrant is the last quadrant before a full rotation to 360°, so any angle in this range belongs here.

Key Concepts

Angle classification

In trigonometry, angles are classified based on their terminal side's location after a rotation from the initial side. This helps in identifying their properties and simplifies calculations. Here's how angles are typically classified:

• Acute angles: These angles are between 0° and 90° and are often seen in the first quadrant.
• Right angles: These are exactly 90° and lie along the positive or negative y-axis.
• Obtuse angles: These lie between greater than 90° but less than 180°, usually falling into the second quadrant.
• Straight angles: A straight angle is 180°, creating a straight line, thus lying on the negative x-axis.
• Reflex angles: If an angle measures between 180° and 360°, it is a reflex angle, often falling into the third or fourth quadrant.
• Full rotation: A 360° angle completes a full circle and returns to the initial position.

Understanding these classifications sets the foundation for further trigonometric calculations and applications.

Quadrant identification

The concept of quadrants is essential in trigonometry as they help to determine the sign and value relationship of trigonometric functions. The Cartesian coordinate system divides the plane into four distinct quadrants. Each quadrant has specific features:

• Quadrant I (0° to 90°): This is the first quadrant where both x and y coordinates are positive. Angles like 31° lie here.
• Quadrant II (90° to 180°): Here, x is negative, and y is positive, resulting in angles reflecting this sign difference.
• Quadrant III (180° to 270°): In this quadrant, both x and y are negative, affecting the sign of trigonometric functions.
• Quadrant IV (270° to 360°): Now, x is positive, and y is negative. An angle like 310° falls into this quadrant.

Identifying which quadrant an angle lies in helps determine the angle's properties, such as sign of sin, cos and tan values.

Standard position

An angle is in standard position when it's drawn on a coordinate plane with its vertex at the origin and its initial side along the positive x-axis. This standardization makes it easier to compare and compute angles since it provides a consistent reference point. The terminal side is then moved counterclockwise or clockwise, depending on whether the angle is positive or negative, respectively. For example:

• A 31° angle starts from the positive x-axis and moves counterclockwise into the first quadrant.
• A 310° angle also starts at the same position but continues counterclockwise through full rotations and eventually stops in the fourth quadrant.

Standard position is essential for creating a uniform framework for multiple trigonometric operations and eases the process of angle measurements.