Find the reference angle in radians for a rotation of 8pi/7.

1. Understand the Concept of Reference Angle: The reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It is always positive and less than or equal to $\frac{\pi}{2}$ radians (or 90 degrees).

1. Analyze the Given Angle: The given angle is $\frac{8\pi}{7}$ radians. This angle is greater than $\pi$ (180 degrees) and less than $\frac{3\pi}{2}$ (270 degrees), which means it lies in the third quadrant.

1. Determine the Quadrant: Since $\pi < \frac{8\pi}{7} < \frac{3\pi}{2}$, the angle is in the third quadrant.

4. Calculate the Reference Angle: In the third quadrant, the reference angle is found by subtracting $\pi$ from the given angle.
Reference Angle = Given Angle - $\pi$
Reference Angle = $\frac{8\pi}{7} - \pi$

5. Perform the Subtraction: To subtract, find a common denominator:
Reference Angle = $\frac{8\pi}{7} - \frac{7\pi}{7}$
Reference Angle = $\frac{8\pi - 7\pi}{7}$
Reference Angle = $\frac{\pi}{7}$

1. Verify the Result: The calculated reference angle is $\frac{\pi}{7}$. This is a positive angle and it is less than $\frac{\pi}{2}$, so it is an acute angle. Therefore, it is a valid reference angle.