Question

draw the angle in standard position. state the quadrant in which the angle lies. work the exercise without converting to degrees.

$\frac{4pi}{3}$

choose the correct graph below.

a.
b.
c.
d.

Explanation:

Step1: Recall quadrant - angle ranges

The first - quadrant angles in radians are in the range \(0\lt\theta\lt\frac{\pi}{2}\), the second - quadrant angles are in the range \(\frac{\pi}{2}\lt\theta\lt\pi\), the third - quadrant angles are in the range \(\pi\lt\theta\lt\frac{3\pi}{2}\), and the fourth - quadrant angles are in the range \(\frac{3\pi}{2}\lt\theta\lt2\pi\).

Step2: Compare the given angle

We are given the angle \(\theta=\frac{4\pi}{3}\). Since \(\pi=\frac{3\pi}{3}\) and \(\frac{4\pi}{3}\), and \(\pi\lt\frac{4\pi}{3}\lt\frac{3\pi}{2}\), the angle \(\frac{4\pi}{3}\) lies in the third quadrant.
In standard position, an angle \(\theta\) is drawn with its vertex at the origin and its initial side along the positive \(x\) - axis. For an angle \(\frac{4\pi}{3}\), we rotate counter - clockwise from the positive \(x\) - axis.

Answer:

The angle \(\frac{4\pi}{3}\) lies in the third quadrant. Without seeing the actual details of the graphs (but based on the fact that for a third - quadrant angle, the terminal side is in the third quadrant), we know that the graph with the terminal side of the angle in the third quadrant is the correct one. If we assume the standard orientation of the graphs with the positive \(x\) - axis to the right and positive \(y\) - axis up, the graph with the terminal side of the angle below the negative \(x\) - axis and to the left of the negative \(y\) - axis is the correct choice.