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.
_ 7𝜋4

Verified step by step guidance

1. Identify the given angle: \( \frac{7\pi}{4} \).
2. Recognize that \( 2\pi \) is a full circle, so \( \frac{7\pi}{4} \) is less than \( 2\pi \) but more than \( \pi \).
3. Since \( \frac{7\pi}{4} \) is more than \( \frac{3\pi}{2} \) (which is \( 270^\circ \)), it lies in the fourth quadrant.
4. Draw the angle in standard position starting from the positive x-axis and moving counterclockwise.
5. Conclude that the terminal side of the angle \( \frac{7\pi}{4} \) ends in the fourth quadrant.

Key Concepts

Standard Position of an Angle

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 2π radians, it can be reduced by subtracting full rotations (2π) to find its equivalent angle in standard position.

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 is where both x and y are positive, Quadrant II has a negative x and positive y, Quadrant III has both negative x and y, and Quadrant IV has a positive x and negative y. Understanding which quadrant an angle lies in is crucial for determining the signs of trigonometric functions.

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 important to visualize their position on the unit circle to determine their corresponding coordinates and the quadrant they occupy.