In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

420°

Verified step by step guidance

1. Step 1: Understand that an angle in standard position starts from the positive x-axis and rotates counterclockwise for positive angles.
2. Step 2: Since the angle given is 420°, recognize that this is more than one full rotation (360°). To find the equivalent angle within one rotation, subtract 360° from 420°: \$420° - 360° = 60°$.
3. Step 3: Draw the angle of 60° starting from the positive x-axis and rotating counterclockwise. This will place the terminal side of the angle in the first quadrant.
4. Step 4: Identify the quadrant where the terminal side lies. Since 60° is between 0° and 90°, the angle lies in the first quadrant.
5. Step 5: Conclude that the angle 420° is coterminal with 60°, and its terminal side lies in the first quadrant.

Key Concepts

Angles in Standard Position

An angle is in standard position when its vertex is at the origin of the coordinate system, and its initial side lies along the positive x-axis. The terminal side is determined by rotating the initial side counterclockwise for positive angles and clockwise for negative angles.

Coterminal Angles and Angle Reduction

Angles that differ by full rotations (360° or 2π radians) share the same terminal side and are called coterminal. To find the quadrant of an angle greater than 360°, subtract multiples of 360° until the angle lies between 0° and 360°, simplifying the analysis.

Quadrants in the Coordinate Plane

The coordinate plane is divided into four quadrants: I (0° to 90°), II (90° to 180°), III (180° to 270°), and IV (270° to 360°). The quadrant where the terminal side of an angle lies helps determine the sign of trigonometric functions and the angle's position.