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/6

Verified step by step guidance

1. Understand that the angle \(\frac{5\pi}{6}\) is given in radians and is in standard position, meaning its vertex is at the origin and its initial side lies along the positive x-axis.
2. Recall that \(\pi\) radians correspond to 180 degrees, so \(\frac{5\pi}{6}\) is slightly less than \(\pi\) (180 degrees), specifically \(\frac{5}{6}\) of \(\pi\).
3. To draw the angle, start from the positive x-axis and rotate counterclockwise by \(\frac{5\pi}{6}\) radians. This rotation will place the terminal side of the angle in the second quadrant.
4. Mark the terminal side of the angle on the circle, which will be between \(\frac{\pi}{2}\) (90 degrees) and \(\pi\) (180 degrees), confirming it lies in the second quadrant.
5. State that the angle \(\frac{5\pi}{6}\) lies in the second quadrant based on its position after rotation.

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 of Angles

Radians measure angles based on the radius of a circle, where 2π radians equal 360 degrees. This unit allows direct use of the circle's arc length and radius, making it natural for trigonometric calculations without converting to degrees.

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants, numbered counterclockwise starting from the upper right. The quadrant in which an angle's terminal side lies is determined by the angle's measure, helping to identify the sign of trigonometric functions.