Problem 45 - Basic Technical Mathematics with Calculus

Question

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

\(31^{\circ}\) is in Quadrant I, \(310^{\circ}\) is in Quadrant IV.

Step 1: Understand Quadrants

In trigonometry, a circle is divided into four quadrants. Quadrant I covers angles from \(0^{\circ}\) to \(90^{\circ}\), Quadrant II from \(90^{\circ}\) to \(180^{\circ}\), Quadrant III from \(180^{\circ}\) to \(270^{\circ}\), and Quadrant IV from \(270^{\circ}\) to \(360^{\circ}\). An angle whose terminal side falls exactly on the horizontal or vertical axis is called a quadrantal angle.

Step 2: Determine the Quadrant for \(31^{\circ}\)

Since \(31^{\circ}\) is greater than \(0^{\circ}\) and less than \(90^{\circ}\), it falls in Quadrant I.

Step 3: Determine the Quadrant for \(310^{\circ}\)

Since \(310^{\circ}\) is greater than \(270^{\circ}\) but less than \(360^{\circ}\), it falls in Quadrant IV.

Key Concepts

Angles

Angles are fundamental in trigonometry, as they serve as the basis for defining the functions that describe various properties of triangles and circles. An angle is formed by the intersection of two rays, sharing a common endpoint called the vertex. In geometry, angles are generally measured in degrees, with a full circle equivalent to 360 degrees.

Understanding angles involves knowing that:

• An acute angle is less than 90 degrees.
• A right angle is exactly 90 degrees.
• An obtuse angle is more than 90 degrees but less than 180 degrees.
• A straight angle is exactly 180 degrees.

The direction of measurement can be clockwise or counterclockwise, affecting the sign of the angle in trigonometric functions. Typically, counterclockwise is taken as the positive direction.

Quadrants

In trigonometry, a coordinate plane is divided into four specific regions known as quadrants. These quadrants help to determine the attributes and signs of trigonometric functions. When plotting angles, the plane starting from the positive x-axis and moving counterclockwise is broken down as follows:

• Quadrant I: Ranges from 0 to 90 degrees, where all trigonometric functions are positive.
• Quadrant II: Ranges from 90 to 180 degrees, where sine is positive and cosine and tangent are negative.
• Quadrant III: Ranges from 180 to 270 degrees, where tangent is positive and sine and cosine are negative.
• Quadrant IV: Ranges from 270 to 360 degrees, where cosine is positive and sine and tangent are negative.

This structure allows for systematic identification and calculation of angles and their trigonometric values.

Quadrantal Angle

A quadrantal angle is a bit unique as it does not lie within any of the four quadrants. Instead, these angles occur precisely on the axes, dividing the plane into its four quadrants. Quadrantal angles always have their terminal side coinciding with one of the coordinate axes:

• At 0 or 360 degrees on the positive x-axis
• At 90 degrees on the positive y-axis
• At 180 degrees on the negative x-axis
• At 270 degrees on the negative y-axis

These angles have significant importance in trigonometry because the values of sine, cosine, and tangent are either 0, 1, or undefined, simplifying many computations.