Problem 57 - Precalculus

Question

State in which quadrant or on which axis each angle with the given measure in standard position would lie. $$145^{\circ}$$

Step-by-Step Solution

Answer

The angle 145° lies in Quadrant II.

Step 1: Understanding Angle Measures and Quadrants

In standard position, an angle is drawn on the coordinate plane with its vertex at the origin and one side along the positive x-axis. Angles are measured in a counterclockwise direction from the positive x-axis. A full circle is divided into 360 degrees, and each quadrant contains 90 degrees. Thus, Quadrant I ranges from 0° to 90°, Quadrant II from 90° to 180°, Quadrant III from 180° to 270°, and Quadrant IV from 270° to 360°.

Step 2: Locate Angle in Quadrants

The angle given is 145°, which we need to place in one of the quadrants. Since Quadrant II covers angles from 90° to 180°, the angle 145° lies within this range. Hence, it is situated in Quadrant II.

Key Concepts

Standard Position

When discussing angles in geometry, particularly on the coordinate plane, the concept of "standard position" is fundamental. An angle is said to be in standard position if its vertex is located at the origin of the coordinate plane, which is the point (0, 0). The initial side of the angle lies along the positive x-axis.
This setup allows for consistent measurement of angles. Angles in standard position are measured from the initial side to the terminal side. The direction of measurement is pivotal: we measure counterclockwise for positive angles and clockwise for negative angles.
This consistent starting point aids in easily comparing angle sizes and locating them accurately on the coordinate plane.

Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is identified using a pair of numerical coordinates. These coordinates are formed by horizontal and vertical lines called axes. The horizontal line is known as the x-axis, and the vertical line is the y-axis.
Together, these axes divide the plane into four equal sections called quadrants. They also meet at the origin (0, 0), which is the central point where both axes intersect. This system helps in plotting points and describing shapes accurately.

• The x-axis is the baseline for measuring angles in standard position.
• The y-axis assists in defining directions of the quadrants.
• Both axes act as a guide to place angles correctly as they provide a precise positional framework.

Understanding the coordinate plane is essential for visualizing and solving problems involving geometric figures and angles.

Quadrants

Quadrants are the four sections on the coordinate plane created by the intersection of the x and y axes. They are numbered counterclockwise, starting from the "northeast" section, which is Quadrant I, and proceeding to Quadrant II, Quadrant III, and Quadrant IV.
This numbering system is used because angles in standard position are also measured counterclockwise. Here's a quick breakdown of each quadrant:

• Quadrant I: Ranges from 0° to 90°. Here both x and y coordinates are positive.
• Quadrant II: Ranges from 90° to 180°. In this quadrant, x is negative while y remains positive.
• Quadrant III: Covers 180° to 270°. Both x and y coordinates are negative.
• Quadrant IV: Spans from 270° to 360°. Here, x is positive and y is negative.

Each quadrant designates a specific combination of signs for x and y coordinates, which helps in understanding the directional placement of the angles and the signs of coordinates for points located within those angles.