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.

Learn how to find the reference angle for -8π/9 in this trigonometry problem. Explore key concepts like coterminal angles, reference angles, and working with radians in different quadrants. This detailed guide is suitable for students in Grades 10-12.

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