Problem 46 - 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. $$180^{\circ}, 92^{\circ}$$

Step-by-Step Solution

Answer

$180^{\circ}$ is a quadrantal angle, and $92^{\circ}$ is in Quadrant II.

Step 1: Understanding Quadrants

In the coordinate plane, the four quadrants are defined by the positive and negative x and y axes. Starting from the positive x-axis and moving counter-clockwise, the quadrants are labeled I, II, III, and IV.

Step 2: Angle in Standard Position

An angle in standard position has its vertex at the origin of the coordinate plane and its initial side along the positive x-axis. The terminal side is determined by the angle's measure, moving counterclockwise for positive angles.

Step 3: Determine Quadrant for $180^{\circ}$

The angle \(180^{\circ}\) lies exactly along the negative x-axis. Angles that lie directly on any axis are called quadrantal angles. Therefore, \(180^{\circ}\) is a quadrantal angle.

Step 4: Determine Quadrant for $92^{\circ}$

For \(92^{\circ}\), we start from the positive x-axis and move counterclockwise. Since \(90^{\circ}\) marks the positive y-axis, \(92^{\circ}\) is slightly past this point and lies in the second quadrant.

Key Concepts

Quadrants in Mathematics

In mathematics, the coordinate plane is a flat surface with two intersecting lines, typically called the x-axis and the y-axis. These lines divide the plane into four regions, or quadrants.

• Quadrant I is where both x and y are positive.
• Quadrant II is where x is negative and y is positive.
• Quadrant III is where both x and y are negative.
• Quadrant IV is where x is positive and y is negative.

When referring to angles in these quadrants, we start from the positive x-axis and move counterclockwise. This is important because the location of the angle in a quadrant helps us understand its direction and its mathematical properties in trigonometry and geometry.

Standard Position Angles

Angles in the coordinate plane are often described by their position relative to the axes. A standard position angle is a way to represent angles starting from a common point.The initial side of a standard position angle always starts on the positive x-axis. From here, the angle opens counterclockwise, a direction usually associated with positive angle values. Conversely, a clockwise direction represents negative angles. Knowing the standard position of an angle helps in determining the quadrant in which the angle's terminal side - or final position - lies. This is helpful in various mathematical calculations and for visualizing problems in trigonometry. For example, an angle of \(92^{\circ}\) from the positive x-axis moves counterclockwise slightly more than \(90^{\circ}\), placing it in the second quadrant.

Quadrantal Angles

Quadrantal angles are a special category in trigonometry and are quite fascinating because of their exact positioning on the axes. An angle that lies directly on any of the coordinate axes - either x or y - is termed a quadrantal angle.The common quadrantal angles are:

• \(0^{\circ}\), or \(360^{\circ}\), on the positive x-axis.
• \(90^{\circ}\) on the positive y-axis.
• \(180^{\circ}\) on the negative x-axis.
• \(270^{\circ}\) on the negative y-axis.

Angles like \(180^{\circ}\) don't fall into any quadrant but lie precisely on an axis. These specific angles play a big role in simplifying trigonometric problems because of the unique values of sine, cosine, and other trigonometric functions at these angles.