text: # Find the value of
  
  Question:
  
  Find the value of $\cos ^{-1}\left(\cos \frac{15 \pi}{6}\right)$
  
  Solution:
  
  We know that $\cos ^{-1}(\cos x)=x$ if $x \in[0, \pi]$, which is the principal value branch of $\cos ^{-1} x$.
  
  Here, $\frac{13 \pi}{6} \notin[0, \pi]$.
  
  Now, $\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)$ can be written as:
  
  $\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)=\cos ^{-1}\left[\cos \left(2 \pi+\frac{\pi}{6}\right)\right]=\cos ^{-1}\left[\cos \left(\frac{\pi}{6}\right)\right]$, where $\frac{\pi}{6} \in[0, \pi]$
  
  $\therefore \cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)=\cos ^{-1}\left[\cos \left(\frac{\pi}{6}\right)\right]=\frac{\pi}{6}$
explanation: The document contains a math question and its corresponding solution. The question asks for the value of \(\cos^{-1}\left(\cos \frac{15 \pi}{6}\right)\), and the solution provides a step-by-step explanation to find the answer, which is \(\frac{\pi}{6}\).
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Find the value of
  
  Question:
  
  Find the value of $\cos ^{-1}\left(\cos \frac{15 \pi}{6}\right)$
  
  Solution:
  
  We know that $\cos ^{-1}(\cos x)=x$ if $x \in[0, \pi]$, which is the principal value branch of $\cos ^{-1} x$.
  
  Here, $\frac{13 \pi}{6} \notin[0, \pi]$.
  
  Now, $\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)$ can be written as:
  
  $\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)=\cos ^{-1}\left[\cos \left(2 \pi+\frac{\pi}{6}\right)\right]=\cos ^{-1}\left[\cos \left(\frac{\pi}{6}\right)\right]$, where $\frac{\pi}{6} \in[0, \pi]$
  
  $\therefore \cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)=\cos ^{-1}\left[\cos \left(\frac{\pi}{6}\right)\right]=\frac{\pi}{6}$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and its corresponding solution. The question asks for the value of \(\cos^{-1}\left(\cos \frac{15 \pi}{6}\right)\), and the solution provides a step-by-step explanation to find the answer, which is \(\frac{\pi}{6}\).
  
  Label: question-and-answer