Find the reforence angle for the angle \( \frac{19 \pi}{3} \) The reforence angie is (Type you answer in radians. Type an integer or a simplifiod fraction.)

To find the reference angle for the angle \( \frac{19\pi}{3} \), we first need to reduce it to an equivalent angle within the range of \( 0 \) to \( 2\pi \). 1. **Find the equivalent angle**: We can do this by subtracting \( 2\pi \) (or \( \frac{6\pi}{3} \)) until the angle is within the desired range. \[ \frac{19\pi}{3} - 2\pi = \frac{19\pi}{3} - \frac{6\pi}{3} = \frac{13\pi}{3} \] \[ \frac{13\pi}{3} - 2\pi = \frac{13\pi}{3} - \frac{6\pi}{3} = \frac{7\pi}{3} \] \[ \frac{7\pi}{3} - 2\pi = \frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3} \] Now, \( \frac{\pi}{3} \) is within the range \( 0 \) to \( 2\pi \). 2. **Determine the reference angle**: The reference angle is the acute angle that the terminal side of the given angle makes with the x-axis. Since \( \frac{\pi}{3} \) is already in the first quadrant, the reference angle is simply: \[ \frac{\pi}{3} \] Thus, the reference angle for \( \frac{19\pi}{3} \) is \[ \boxed{\frac{\pi}{3}} \]