text: # Find the principal value of each of the following:
  
  Question:
  
  Find the principal value of each of the following:
  
  (i) $\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)$
  
  (ii) $\sin ^{-1}\left(\cos \frac{2 \pi}{3}\right)$
  
  (iii) $\sin ^{-1}\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)$
  
  (iv) $\sin ^{-1}\left(\frac{\sqrt{3}+1}{2 \sqrt{2}}\right)$
  
  (v) $\sin ^{-1}\left(\cos \frac{3 \pi}{4}\right)$
  
  (vi) $\sin ^{-1}\left(\tan \frac{5 \pi}{4}\right)$
  
  Solution:
  
  (i) $\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)=\sin ^{-1}\left[\sin \left(-\frac{\pi}{3}\right)\right]=-\frac{\pi}{3}$
  
  (ii) $\sin ^{-1}\left(\cos \frac{2 \pi}{3}\right)=\sin ^{-1}\left(-\frac{1}{2}\right)=\sin ^{-1}\left[\sin \left(-\frac{\pi}{6}\right)\right]=-\frac{\pi}{6}$
  
  (iii) $\sin ^{-1}\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)=\sin ^{-1}\left(\sin \frac{\pi}{12}\right)=\frac{\pi}{12}$
  
  (iv) $\sin ^{-1}\left(\frac{\sqrt{3}+1}{2 \sqrt{2}}\right)=\sin ^{-1}\left(\sin \frac{5 \pi}{12}\right)=\frac{5 \pi}{12}$
  
  (v) $\sin ^{-1}\left(\cos \frac{3 \pi}{4}\right)=\sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right)=\sin ^{-1}\left[\sin \left(-\frac{\pi}{4}\right)\right]=-\frac{\pi}{4}$
  
  (vi) $\sin ^{-1}\left(\tan \frac{5 \pi}{4}\right)=\sin ^{-1}(1)=\sin ^{-1}\left[\sin \left(\frac{\pi}{2}\right)\right]=\frac{\pi}{2}$
explanation: The document contains a series of math questions followed by their respective solutions. Each question asks for the principal value of a given inverse sine function, and the solutions provide the step-by-step evaluation of these functions.
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 principal value of each of the following:
  
  Question:
  
  Find the principal value of each of the following:
  
  (i) $\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)$
  
  (ii) $\sin ^{-1}\left(\cos \frac{2 \pi}{3}\right)$
  
  (iii) $\sin ^{-1}\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)$
  
  (iv) $\sin ^{-1}\left(\frac{\sqrt{3}+1}{2 \sqrt{2}}\right)$
  
  (v) $\sin ^{-1}\left(\cos \frac{3 \pi}{4}\right)$
  
  (vi) $\sin ^{-1}\left(\tan \frac{5 \pi}{4}\right)$
  
  Solution:
  
  (i) $\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)=\sin ^{-1}\left[\sin \left(-\frac{\pi}{3}\right)\right]=-\frac{\pi}{3}$
  
  (ii) $\sin ^{-1}\left(\cos \frac{2 \pi}{3}\right)=\sin ^{-1}\left(-\frac{1}{2}\right)=\sin ^{-1}\left[\sin \left(-\frac{\pi}{6}\right)\right]=-\frac{\pi}{6}$
  
  (iii) $\sin ^{-1}\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)=\sin ^{-1}\left(\sin \frac{\pi}{12}\right)=\frac{\pi}{12}$
  
  (iv) $\sin ^{-1}\left(\frac{\sqrt{3}+1}{2 \sqrt{2}}\right)=\sin ^{-1}\left(\sin \frac{5 \pi}{12}\right)=\frac{5 \pi}{12}$
  
  (v) $\sin ^{-1}\left(\cos \frac{3 \pi}{4}\right)=\sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right)=\sin ^{-1}\left[\sin \left(-\frac{\pi}{4}\right)\right]=-\frac{\pi}{4}$
  
  (vi) $\sin ^{-1}\left(\tan \frac{5 \pi}{4}\right)=\sin ^{-1}(1)=\sin ^{-1}\left[\sin \left(\frac{\pi}{2}\right)\right]=\frac{\pi}{2}$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a series of math questions followed by their respective solutions. Each question asks for the principal value of a given inverse sine function, and the solutions provide the step-by-step evaluation of these functions.
  
  Label: multiple-question-and-answer