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.
14𝜋3

Verified step by step guidance

1. Identify the given angle: \( \frac{14\pi}{3} \).
2. Recognize that a full circle is \( 2\pi \) radians.
3. Divide \( \frac{14\pi}{3} \) by \( 2\pi \) to find how many full circles it contains.
4. Calculate the remainder of \( \frac{14\pi}{3} \) after subtracting the full circles to find the equivalent angle within one circle.
5. Determine the quadrant by locating the equivalent angle on the unit circle.

Key Concepts

Standard Position of Angles

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 is greater than 360 degrees or less than 0 degrees, it can be reduced to an equivalent angle within the range of 0 to 360 degrees by subtracting or adding full rotations.

Quadrants of the Coordinate System

The rectangular coordinate system is divided into four quadrants based on the signs of the x and y coordinates. Quadrant I contains angles from 0 to 90 degrees (or 0 to π/2 radians), Quadrant II from 90 to 180 degrees (or π/2 to π radians), Quadrant III from 180 to 270 degrees (or π to 3π/2 radians), and Quadrant IV from 270 to 360 degrees (or 3π/2 to 2π radians). Identifying the quadrant helps determine the sign of the sine and cosine of the angle.

Radians and Angle Measurement

Radians are a unit of angular measure where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. The full circle is 2π radians, which corresponds to 360 degrees. When working with angles in radians, it is essential to understand how to convert between radians and degrees, although the question specifies to work directly in radians without conversion.