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: \( \frac{3\pi}{4} \).
2. Recognize that \( \frac{3\pi}{4} \) is less than \( \pi \) (which is \( 180^\circ \)), so the angle is in the second quadrant.
3. Draw the initial side of the angle along the positive x-axis.
4. Measure the angle counterclockwise from the positive x-axis to reach \( \frac{3\pi}{4} \).
5. Conclude that the terminal side of the angle \( \frac{3\pi}{4} \) lies in the second 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 measures more than 360 degrees or less than 0 degrees, it can be reduced to an equivalent angle within the range of 0 to 360 degrees.

Quadrants of the Coordinate System

The rectangular coordinate system is divided into four quadrants. Quadrant I is where both x and y are positive, Quadrant II has a negative x and positive y, Quadrant III has both x and y negative, and Quadrant IV has a positive x and negative y. Understanding which quadrant an angle lies in is crucial for determining the signs of the sine and cosine values associated with that angle.

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 quadrant.