text: # What is cos^theta in terms of sintheta?
  
  Dec 18, 2015
  
  If $\theta$ is in Q1 or Q4 then $\cos \theta = \sqrt{1 - {\sin}^{2} \theta}$
  
  If $\theta$ is in Q2 or Q3 then $\cos \theta = - \sqrt{1 - {\sin}^{2} \theta}$
  
  #### Explanation:
  
  We always have ${\cos}^{2} \theta + {\sin}^{2} \theta = 1$ regardless of which quadrant $\theta$ is in. Hence we always have:
  
  $\cos \theta = \pm \sqrt{1 - {\sin}^{2} \theta}$
  
  If $\theta$ is in Q1 or Q4 then $\cos \theta \ge 0$, hence:
  
  $\cos \theta = \sqrt{1 - {\sin}^{2} \theta}$
  
  If $\theta$ is in Q2 or Q3 then $\cos \theta \le 0$, hence:
  
  $\cos \theta = - \sqrt{1 - {\sin}^{2} \theta}$
explanation: The document contains a math question and a detailed answer. The question asks for the expression of \(\cos \theta\) in terms of \(\sin \theta\), and the answer provides the solution based on the quadrant in which \(\theta\) lies.
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: # What is cos^theta in terms of sintheta?
  
  Dec 18, 2015
  
  If $\theta$ is in Q1 or Q4 then $\cos \theta = \sqrt{1 - {\sin}^{2} \theta}$
  
  If $\theta$ is in Q2 or Q3 then $\cos \theta = - \sqrt{1 - {\sin}^{2} \theta}$
  
  #### Explanation:
  
  We always have ${\cos}^{2} \theta + {\sin}^{2} \theta = 1$ regardless of which quadrant $\theta$ is in. Hence we always have:
  
  $\cos \theta = \pm \sqrt{1 - {\sin}^{2} \theta}$
  
  If $\theta$ is in Q1 or Q4 then $\cos \theta \ge 0$, hence:
  
  $\cos \theta = \sqrt{1 - {\sin}^{2} \theta}$
  
  If $\theta$ is in Q2 or Q3 then $\cos \theta \le 0$, hence:
  
  $\cos \theta = - \sqrt{1 - {\sin}^{2} \theta}$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a detailed answer. The question asks for the expression of \(\cos \theta\) in terms of \(\sin \theta\), and the answer provides the solution based on the quadrant in which \(\theta\) lies.
  
  Label: question-and-answer