Work each problem.


Consider each angle in standard position having the given radian measure. In what quadrant does the terminal side lie?


3

Verified step by step guidance

1. Recall that the quadrants in the coordinate plane are divided based on the angle's measure in radians: Quadrant I is from \(0\) to \(\frac{\pi}{2}\), Quadrant II is from \(\frac{\pi}{2}\) to \(\pi\), Quadrant III is from \(\pi\) to \(\frac{3\pi}{2}\), and Quadrant IV is from \(\frac{3\pi}{2}\) to \(2\pi\).
2. Identify the approximate value of the given angle \(3\) radians in terms of \(\pi\). Since \(\pi \approx 3.1416\), the angle \(3\) radians is slightly less than \(\pi\).
3. Compare the angle \(3\) radians to the quadrant boundaries: since \(3\) is greater than \(\frac{\pi}{2} \approx 1.5708\) and less than \(\pi \approx 3.1416\), the angle lies between \(\frac{\pi}{2}\) and \(\pi\).
4. Conclude that the terminal side of the angle \(3\) radians lies in Quadrant II because it is between \(\frac{\pi}{2}\) and \(\pi\).
5. Remember that angles in standard position start from the positive x-axis and rotate counterclockwise, so locating the quadrant depends on where the angle measure falls within these intervals.

Key Concepts

Radian Measure and Angle Conversion

Radian measure expresses angles based on the radius of a circle, where 2π radians equal 360 degrees. Understanding how to interpret and convert radians helps in identifying the angle's position on the coordinate plane.

Standard Position of an Angle

An angle in standard position has its vertex at the origin and its initial side along the positive x-axis. The terminal side rotates counterclockwise for positive angles, determining the angle's location in one of the four quadrants.

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants, each spanning π/2 radians (90 degrees). Knowing the radian ranges for each quadrant allows you to identify where the terminal side of an angle lies based on its radian measure.