text: ## Trigonometry (10th Edition)
  
  $6i$
  First, we use the product theorem to multiply the absolute values and add the arguments: $(\sqrt{6} cis 120^{\circ})(\sqrt{6} cis (-30^{\circ})) \\=\sqrt{6}\sqrt{6} cis (120^{\circ}-30^{\circ}) \\=6 cis (90^{\circ})$ Next, we change the expression into its equivalent form: $=6 cis (90^{\circ}) \\6(\cos90^{\circ}+i\sin90^{\circ})$ Since we know that $\cos90^{\circ}=0$ and $\sin90^{\circ}=1$, we substitute these values in the expression and simplify: $6(\cos90^{\circ}+i\sin90^{\circ}) \\=6(i) \\=6i$
explanation: The document contains a detailed solution to a trigonometry problem involving the multiplication of complex numbers in polar form. It shows the steps to multiply the magnitudes, add the angles, and convert the result back to rectangular form. The final answer is provided as \(6i\).
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: ## Trigonometry (10th Edition)
  
  $6i$
  First, we use the product theorem to multiply the absolute values and add the arguments: $(\sqrt{6} cis 120^{\circ})(\sqrt{6} cis (-30^{\circ})) \\=\sqrt{6}\sqrt{6} cis (120^{\circ}-30^{\circ}) \\=6 cis (90^{\circ})$ Next, we change the expression into its equivalent form: $=6 cis (90^{\circ}) \\6(\cos90^{\circ}+i\sin90^{\circ})$ Since we know that $\cos90^{\circ}=0$ and $\sin90^{\circ}=1$, we substitute these values in the expression and simplify: $6(\cos90^{\circ}+i\sin90^{\circ}) \\=6(i) \\=6i$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a detailed solution to a trigonometry problem involving the multiplication of complex numbers in polar form. It shows the steps to multiply the magnitudes, add the angles, and convert the result back to rectangular form. The final answer is provided as \(6i\).
  
  Label: question-and-answer