To find the reference angle for $-\frac{8\pi}{9}$, follow these steps:

Step 1: Determine the angle's equivalent in the positive direction

Since the angle is negative, we first convert it to a positive equivalent by adding $2\pi$ (a full circle in radians) until the angle falls within the standard $0 \leq \theta < 2\pi$ range:

$-\frac{8\pi}{9} + 2\pi = -\frac{8\pi}{9} + \frac{18\pi}{9} = \frac{10\pi}{9}$

Thus, $\frac{10\pi}{9}$ is the coterminal positive angle.

Step 2: Identify the quadrant of the angle

Since $\frac{10\pi}{9}$ is between $\pi$ and $2\pi$ (since $\pi = \frac{9\pi}{9}$ and $2\pi = \frac{18\pi}{9}$), the angle is located in the third quadrant.

Step 3: Find the reference angle

The reference angle is the acute angle formed with the x-axis. In the third quadrant, the reference angle is calculated by subtracting $\pi$ from the given angle:

$\text{Reference Angle} = \frac{10\pi}{9} - \pi = \frac{10\pi}{9} - \frac{9\pi}{9} = \frac{\pi}{9}$

Thus, the reference angle for $-\frac{8\pi}{9}$ is $\frac{\pi}{9}$.

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Would you like more details on reference angles or have any questions?

Here are 5 related questions to explore:

1. What is the reference angle for $\frac{7\pi}{6}$?
2. How do you convert an angle from degrees to radians?
3. What is a coterminal angle and how is it calculated?
4. How do you determine in which quadrant an angle lies?
5. How do reference angles help in evaluating trigonometric functions?

Tip: When working with radians, remember that $2\pi$ represents a full circle, so adjusting angles by adding or subtracting $2\pi$ can simplify negative or large angles.