Work each problem.


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


4

Verified step by step guidance

1. Recall that the standard position of an angle means its vertex is at the origin and its initial side lies along the positive x-axis. The terminal side is the ray obtained after rotating the initial side by the given angle measure.
2. Understand that the radian measure of an angle corresponds to the length of the arc on the unit circle subtended by that angle. The full circle corresponds to \(2\pi\) radians, which is approximately 6.283 radians.
3. Identify the quadrants based on radian measures: 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 \(2\pi\).
4. Compare the given angle measure, which is 4 radians, to these quadrant boundaries. Since \(\pi \approx 3.1416\) and \(\frac{3\pi}{2} \approx 4.7124\), 4 radians lies between \(\pi\) and \(\frac{3\pi}{2}\).
5. Conclude that because 4 radians is between \(\pi\) and \(\frac{3\pi}{2}\), the terminal side of the angle lies in Quadrant III.

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 determine 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, and its position determines the quadrant.

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants, each corresponding to specific angle ranges in radians: Quadrant I (0 to π/2), II (π/2 to π), III (π to 3π/2), and IV (3π/2 to 2π). Identifying the quadrant requires comparing the angle to these ranges.