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.

-5𝜋/4

Verified step by step guidance

1. Identify that the angle given is \(\frac{5\pi}{4}\) radians, which is in standard position, meaning it starts from the positive x-axis and rotates counterclockwise.
2. Recall that one full rotation around the circle is \(2\pi\) radians, and the circle is divided into four quadrants, each spanning \(\frac{\pi}{2}\) radians.
3. Calculate the approximate position of \(\frac{5\pi}{4}\) by noting that \(\frac{4\pi}{4} = \pi\) radians corresponds to the negative x-axis, so \(\frac{5\pi}{4}\) is \(\frac{\pi}{4}\) radians past \(\pi\).
4. Determine that \(\frac{5\pi}{4}\) lies in the third quadrant because it is between \(\pi\) and \(\frac{3\pi}{2}\) radians.
5. To draw the angle, start from the positive x-axis, rotate counterclockwise past \(\pi\) (180 degrees), and continue \(\frac{\pi}{4}\) radians (45 degrees) into the third 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 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. One full rotation around the circle equals 2π radians. The circle is 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: I (x>0, y>0), II (x<0, y>0), III (x<0, y<0), and IV (x>0, y<0). The quadrant in which an angle's terminal side lies is determined by the angle's radian measure and helps in understanding the sign of trigonometric functions.