Question

choose the figure that shows an angle of $\frac{19pi}{6}$ radians in standard position, then state in which quadrant the angle lies. choose the correct answer below.

Explanation:

Step1: Simplify the angle

We know that \(2\pi\) radians is one - full rotation. We can rewrite \(\frac{19\pi}{6}\) as \(\frac{12\pi + 7\pi}{6}=2\pi+\frac{7\pi}{6}\). Since adding \(2\pi\) to an angle does not change its position in the coordinate - plane, we can focus on \(\frac{7\pi}{6}\).

Step2: Determine the quadrant

We know that \(\pi<\frac{7\pi}{6}<\frac{3\pi}{2}\). Angles between \(\pi\) and \(\frac{3\pi}{2}\) radians lie in the third quadrant.

Answer:

The angle \(\frac{19\pi}{6}\) lies in the third quadrant. (Since the question only asks for the quadrant and not for the figure selection as the figures are not described in detail, we focus on the quadrant determination. If the figures were described, we would match the figure with the correct angle orientation in the third - quadrant).