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.

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Verified step by step guidance

1. Identify the angle given in radians: \(\frac{2\pi}{3}\).
2. Recall that angles in standard position start from the positive x-axis and rotate counterclockwise.
3. Since \(2\pi\) radians correspond to a full circle (360 degrees), \(\frac{2\pi}{3}\) radians is two-thirds of \(\pi\) radians (180 degrees).
4. Locate \(\frac{2\pi}{3}\) on the unit circle by dividing the circle into three equal parts, each part being \(\frac{2\pi}{3}\) radians. This angle lies between \(\frac{\pi}{2}\) and \(\pi\) radians.
5. Determine the quadrant: Since \(\frac{2\pi}{3}\) radians is greater than \(\frac{\pi}{2}\) but less than \(\pi\), the terminal side of the angle lies in the second 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 then determined by rotating the initial side counterclockwise for positive angles or clockwise for negative angles.

Radian Measure

Radian measure defines angles based on the radius of a circle. One radian is the angle subtended by an arc equal in length to the radius. Since the circumference of a circle is 2π times the radius, a full rotation corresponds to 2π radians.

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants by the x- and y-axes. The quadrant in which an angle's terminal side lies depends on the angle's measure. Knowing the quadrant helps determine the sign of trigonometric functions and the angle's position.