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°

Key Concepts

Standard Position of an Angle

An angle is said to be 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 angle is measured counterclockwise from the initial side. If the angle exceeds 360 degrees, it can be represented by subtracting 360 degrees until it falls within the range of 0 to 360 degrees.

Quadrants of the Coordinate System

The rectangular coordinate system is divided into four quadrants. Quadrant I is where both x and y are positive, Quadrant II has a negative x and positive y, Quadrant III has both negative x and y, and Quadrant IV has a positive x and negative y. Understanding which quadrant an angle lies in is crucial for determining the sign of the sine and cosine values associated with that angle.

Radians and Degrees

Radians and degrees are two units for measuring angles. One complete revolution (360 degrees) is equivalent to 2π radians. When working with angles, especially in trigonometry, it is often necessary to convert between these two units. However, in this exercise, the angle is given in degrees, and the task is to analyze it without converting to radians, which emphasizes understanding the relationship between the two systems.