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.

7𝜋/4

Verified step by step guidance

1. Identify that the angle given is \(\frac{7\pi}{4}\) radians, which is measured in standard position starting from the positive x-axis and moving counterclockwise.
2. Recall that one full revolution around the circle is \(2\pi\) radians, so \(\frac{7\pi}{4}\) is less than \(2\pi\) and lies within one full rotation.
3. Note that the circle is divided into four quadrants, each spanning \(\frac{\pi}{2}\) radians: Quadrant I from \(0\) to \(\frac{\pi}{2}\), Quadrant II from \(\frac{\pi}{2}\) to \(\pi\), Quadrant III from \(\pi\) to \(\frac{3\pi}{2}\), and Quadrant IV from \(\frac{3\pi}{2}\) to \(2\pi\).
4. Since \(\frac{7\pi}{4}\) is between \(\frac{3\pi}{2}\) and \(2\pi\), the terminal side of the angle lies in Quadrant IV.
5. To draw the angle, start at the positive x-axis and rotate counterclockwise through \(\frac{7\pi}{4}\) radians, which corresponds to 3 full \(\frac{\pi}{2}\) quadrants plus an additional \(\frac{\pi}{4}\), ending in Quadrant IV.

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.

Radian Measure and Circle Division

Radian measure relates the length of an arc on a unit circle to the angle it subtends at the center. One full rotation around the circle equals 2π radians. The circle can be divided into quadrants, each spanning π/2 radians, which helps locate the angle's terminal side.

Quadrants in 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.