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To find the reference angle \( \theta' \) for an angle that measures \( \frac{23\pi}{6} \), we can follow these steps:

1. **Convert the angle to an equivalent angle within \( 0 \) to \( 2\pi \)**:
   Since \( \frac{23\pi}{6} \) is greater than \( 2\pi \), we can find an equivalent angle by subtracting \( 2\pi \) until we’re within \( 0 \leq \theta < 2\pi \).
   
   \( 2\pi \) in terms of sixths is \( \frac{12\pi}{6} \), so:
   \[
   \frac{23\pi}{6} - \frac{12\pi}{6} = \frac{11\pi}{6}
   \]
   So, \( \frac{23\pi}{6} \) is equivalent to \( \frac{11\pi}{6} \).

2. **Determine the reference angle \( \theta' \)**:
   The angle \( \frac{11\pi}{6} \) is in the fourth quadrant, where reference angles are calculated as \( 2\pi - \theta \).
   \[
   \theta' = 2\pi - \frac{11\pi}{6} = \frac{12\pi}{6} - \frac{11\pi}{6} = \frac{\pi}{6}
   \]

Therefore, the reference angle \( \theta' \) for \( \frac{23\pi}{6} \) is \( \frac{\pi}{6} \).

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

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Here are some related questions for further practice:

1. How do you find the reference angle for \( \frac{17\pi}{4} \)?
2. What is the reference angle for an angle measuring \( -\frac{7\pi}{3} \)?
3. How can you convert an angle in degrees to a reference angle?
4. What is the reference angle for \( \frac{5\pi}{3} \)?
5. How would you determine the quadrant of an angle given in radians?

**Tip:** Remember that reference angles are always positive and fall between \( 0 \) and \( \frac{\pi}{2} \).

Find the reference angle θ' of an angle that measures 23π/6.

Learn how to calculate the reference angle for an angle measuring 23π/6 in radians. This tutorial covers key trigonometry concepts, including reference angles, quadrant positioning, and angle reduction within a full circle (0 to 2π). Suitable for high school students in Grades 10-12.

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