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.

14𝜋/3

Verified step by step guidance

1. Identify that the angle given is \(\frac{14\pi}{3}\) radians, which is greater than \(2\pi\) radians (one full rotation around the circle).
2. To find the equivalent angle within one full rotation, subtract multiples of \(2\pi\) from \(\frac{14\pi}{3}\) until the result is between \(0\) and \(2\pi\). Use the formula: \(\theta_{equiv} = \theta - 2\pi \times k\), where \(k\) is an integer.
3. Calculate \(k\) by dividing the numerator of the angle by the denominator times 2 (i.e., \(k = \left\lfloor \frac{14}{3 \times 2} \right\rfloor\)), then subtract \(2\pi k\) from \(\frac{14\pi}{3}\) to get the equivalent angle in radians.
4. Once you have the equivalent angle, determine its position on the unit circle by comparing it to the standard quadrant boundaries: Quadrant I (\(0\) to \(\frac{\pi}{2}\)), Quadrant II (\(\frac{\pi}{2}\) to \(\pi\)), Quadrant III (\(\pi\) to \(\frac{3\pi}{2}\)), and Quadrant IV (\(\frac{3\pi}{2}\) to \(2\pi\)).
5. Draw the angle in standard position starting from the positive x-axis and moving counterclockwise by the equivalent angle measure. State the quadrant where the terminal side of the angle lies.

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 2π radians equal one full rotation (360 degrees). To find the position of an angle like 14π/3, you reduce it by subtracting multiples of 2π to find its equivalent angle within one full rotation. This concept is essential for working directly with radians 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 determine the angle's properties and trigonometric function signs. This is crucial for interpreting the angle's position after drawing it in standard position.