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.

16𝜋/3

Verified step by step guidance

1. Step 1: Understand that the angle is given in radians as \(\frac{16\pi}{3}\). Since one full rotation around the circle is \(2\pi\) radians, we need to find the equivalent angle between \(0\) and \(2\pi\) by subtracting multiples of \(2\pi\).
2. Step 2: Calculate how many full rotations are contained in \(\frac{16\pi}{3}\) by dividing \(\frac{16\pi}{3}\) by \(2\pi\): \(\frac{16\pi}{3} \div 2\pi = \frac{16}{3} \times \frac{1}{2} = \frac{8}{3}\). This means the angle makes \(2\) full rotations (since \(\frac{8}{3} = 2 + \frac{2}{3}\)) plus an extra \(\frac{2}{3}\) of a rotation.
3. Step 3: Find the remaining angle after subtracting \(2\) full rotations: \(\frac{16\pi}{3} - 2 \times 2\pi = \frac{16\pi}{3} - 4\pi = \frac{16\pi}{3} - \frac{12\pi}{3} = \frac{4\pi}{3}\). This is the angle in standard position between \(0\) and \(2\pi\).
4. Step 4: Determine the quadrant of the angle \(\frac{4\pi}{3}\). Since \(\pi < \frac{4\pi}{3} < \frac{3\pi}{2}\), the angle lies in the third quadrant.
5. Step 5: To draw the angle, start from the positive x-axis and rotate counterclockwise by \(\frac{4\pi}{3}\) radians, which is \(240^\circ\) (though we do not convert to degrees for the problem). Mark the terminal side of the angle in the third quadrant on the circle.

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. Understanding this helps in accurately drawing and locating angles on the coordinate plane.

Radian Measure and Circle Rotation

Radians measure angles based on the radius of a circle, where one full rotation equals 2π radians. To work with angles larger than 2π, you subtract multiples of 2π to find the equivalent angle within one full rotation. This concept is essential for interpreting and drawing angles like 16π/3 without converting to degrees.

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants, each defined by the signs of x and y coordinates. Knowing which quadrant an angle's terminal side lies in helps describe its position and properties. This is crucial for identifying the quadrant of an angle drawn in standard position.