Determine the quadrant in which the terminal side of \( \frac{6 \pi}{5} \) lies

To determine the quadrant in which the terminal side of \( \frac{6 \pi}{5} \) radians lies, we can first convert the radian measure into degrees. Since \( \pi \) radians is equal to \( 180^\circ \), we calculate: \[ \frac{6 \pi}{5} \cdot \frac{180^\circ}{\pi} = \frac{6 \cdot 180^\circ}{5} = \frac{1080^\circ}{5} = 216^\circ. \] Now, looking at the unit circle, \( 216^\circ \) is greater than \( 180^\circ \) but less than \( 270^\circ \), which places it in the third quadrant. Therefore, the terminal side of \( \frac{6 \pi}{5} \) lies in the third quadrant. Also, it's fascinating to note that angles in the third quadrant have both sine and cosine values negative. This is crucial when solving problems related to trigonometric functions! Isn't geometry amazing? If you're ever stuck on similar questions, remember to visualize the unit circle and break angles down into manageable parts: knowing the rotation can greatly simplify your calculations!