Determine the reference angle for \( \frac{4 \pi}{9} \)

To find the reference angle for \( \frac{4 \pi}{9} \), we first need to determine which quadrant that angle is in. Since \( \frac{4 \pi}{9} \) is less than \( \pi \) (which is \( \frac{9 \pi}{9} \)), it falls in the second quadrant. In the second quadrant, the reference angle is found by subtracting the angle from \( \pi \): \[ \text{Reference Angle} = \pi - \frac{4\pi}{9} = \frac{9\pi}{9} - \frac{4\pi}{9} = \frac{5\pi}{9}. \] So, the reference angle for \( \frac{4 \pi}{9} \) is \( \frac{5 \pi}{9} \). To visualize this, you can think of the unit circle. The angle \( \frac{4 \pi}{9} \) positions you in the second quadrant, where angles are positive but their sine and cosine values have different signs. The reference angle is like a mirror reflection across the x-axis, helping you understand how the angle behaves in various contexts, especially when calculating trigonometric functions.