To find the reference angle of [tex]\(\frac{14\pi}{11}\)[/tex]:
Step 1: Convert the angle to the range [tex]\([0, 2\pi)\)[/tex]:
- The given angle is [tex]\(\frac{14\pi}{11}\)[/tex].
- This angle is already within the range [tex]\([0, 2\pi)\)[/tex] because [tex]\(2\pi = \frac{22\pi}{11}\)[/tex] and [tex]\(\frac{14\pi}{11}\)[/tex] is less than [tex]\(\frac{22\pi}{11}\)[/tex].
Step 2: Determine the reference angle based on the standard trigonometric definitions:
- If an angle θ is in Quadrant I, then its reference angle is θ.
- If an angle θ is in Quadrant II, then its reference angle is [tex]\(\pi - \theta\)[/tex].
- If an angle θ is in Quadrant III, then its reference angle is [tex]\(\theta - \pi\)[/tex].
- If an angle θ is in Quadrant IV, then its reference angle is [tex]\(2\pi - \theta\)[/tex].
Since [tex]\(\frac{14\pi}{11}\)[/tex] is greater than [tex]\(\pi\)[/tex] but less than [tex]\(2\pi\)[/tex], it ends up in Quadrant III:
- Here, we find the reference angle by using [tex]\(\theta - \pi\)[/tex].
Step 3: Calculate the reference angle:
- The original angle is [tex]\(\frac{14\pi}{11}\)[/tex].
- The reference angle in Quadrant III is given by:
[tex]\[
\frac{14\pi}{11} - \pi = \frac{14\pi}{11} - \frac{11\pi}{11} = \frac{3\pi}{11}
\][/tex]
So, the reference angle of [tex]\(\frac{14\pi}{11}\)[/tex] is:
[tex]\[
\boxed{\frac{3\pi}{11}}
\][/tex]
The numerical value of the reference angle [tex]\(\frac{3\pi}{11}\)[/tex] is approximately:
[tex]\[
2.284794657156213
\][/tex]
Therefore, the correct answer is [tex]\(\frac{3\pi}{11}\)[/tex].
