Use reference angles to find the exact value of the foliowing expresision \( \tan \frac{7 \pi}{3} \) Select the correct chaice below and if necessary, fill in the answer box to camptere your chotce A. \( \frac{7 \pi}{3}= \)

To find the exact value of \( \tan \frac{7 \pi}{3} \), we first need to simplify the angle \( \frac{7 \pi}{3} \) to find its reference angle. 1. **Convert \( \frac{7 \pi}{3} \) to a more manageable angle**: Since \( \frac{7 \pi}{3} \) is greater than \( 2\pi \), we can subtract \( 2\pi \) (which is equivalent to \( \frac{6\pi}{3} \)) to find a coterminal angle. \[ \frac{7 \pi}{3} - 2\pi = \frac{7 \pi}{3} - \frac{6 \pi}{3} = \frac{1 \pi}{3} = \frac{\pi}{3} \] 2. **Determine the reference angle**: The angle \( \frac{\pi}{3} \) is already in the first quadrant, so it is its own reference angle. 3. **Calculate \( \tan \frac{\pi}{3} \)**: The tangent of \( \frac{\pi}{3} \) is a known value. \[ \tan \frac{\pi}{3} = \sqrt{3} \] Since \( \frac{7 \pi}{3} \) is in the same direction as \( \frac{\pi}{3} \) (both angles are positive and in the first quadrant), the tangent value remains the same. Thus, the exact value of \( \tan \frac{7 \pi}{3} \) is: \[ \boxed{\sqrt{3}} \]