text: ## Trigonometry (11th Edition) Clone
  
  $$\sin\theta=\frac{\sqrt{2x+1}}{|x+1|}$$
  $$\cos\theta=\frac{x}{x+1}$$ According to the Pythagorean Identity $$\sin^2\theta+\cos^2\theta=1$$ we can rewrite as $$\sin^2\theta=1-\cos^2\theta$$ Then, $$\sin\theta=\sqrt{1-\cos^2\theta}$$ This is the way we would be able to find an expression of $x$ for $\sin\theta$ 1) Find $\cos^2\theta$ $$\cos^2\theta=\frac{x^2}{(x+1)^2}$$ 2) Find $\sin^2\theta$ $$\sin^2\theta=1-\cos^2\theta$$ $$\sin^2\theta=1-\frac{x^2}{(x+1)^2}$$ $$\sin^2\theta=\frac{(x+1)^2-x^2}{(x+1)^2}$$ $$\sin^2\theta=\frac{(x+1-x)(x+1+x)}{(x+1)^2}$$ (for $a^2-b^2=(a-b)(a+b)$) $$\sin^2\theta=\frac{1\times(2x+1)}{(x+1)^2}$$ $$\sin^2\theta=\frac{2x+1}{(x+1)^2}$$ 3) Find $\sin\theta$ $$\sin\theta=\sqrt{\sin^2\theta}$$ $$\sin\theta=\frac{\sqrt{2x+1}}{\sqrt{(x+1)^2}}$$ $$\sin\theta=\frac{\sqrt{2x+1}}{|x+1|}$$ (since $\sqrt{a^2}=|a|$) Unfortunately, we cannot eliminate the absolute value $||$ sign, since there is not enough information to know whether $(x+1)\gt0$ or $\lt0$.
explanation: The document is a detailed mathematical derivation from a trigonometric identity. It starts with given expressions for \(\sin\theta\) and \(\cos\theta\) and uses the Pythagorean identity to derive an expression for \(\sin\theta\) in terms of \(x\). The document does not contain a specific question or a question and answer format.
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 (11th Edition) Clone
  
  $$\sin\theta=\frac{\sqrt{2x+1}}{|x+1|}$$
  $$\cos\theta=\frac{x}{x+1}$$ According to the Pythagorean Identity $$\sin^2\theta+\cos^2\theta=1$$ we can rewrite as $$\sin^2\theta=1-\cos^2\theta$$ Then, $$\sin\theta=\sqrt{1-\cos^2\theta}$$ This is the way we would be able to find an expression of $x$ for $\sin\theta$ 1) Find $\cos^2\theta$ $$\cos^2\theta=\frac{x^2}{(x+1)^2}$$ 2) Find $\sin^2\theta$ $$\sin^2\theta=1-\cos^2\theta$$ $$\sin^2\theta=1-\frac{x^2}{(x+1)^2}$$ $$\sin^2\theta=\frac{(x+1)^2-x^2}{(x+1)^2}$$ $$\sin^2\theta=\frac{(x+1-x)(x+1+x)}{(x+1)^2}$$ (for $a^2-b^2=(a-b)(a+b)$) $$\sin^2\theta=\frac{1\times(2x+1)}{(x+1)^2}$$ $$\sin^2\theta=\frac{2x+1}{(x+1)^2}$$ 3) Find $\sin\theta$ $$\sin\theta=\sqrt{\sin^2\theta}$$ $$\sin\theta=\frac{\sqrt{2x+1}}{\sqrt{(x+1)^2}}$$ $$\sin\theta=\frac{\sqrt{2x+1}}{|x+1|}$$ (since $\sqrt{a^2}=|a|$) Unfortunately, we cannot eliminate the absolute value $||$ sign, since there is not enough information to know whether $(x+1)\gt0$ or $\lt0$.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a detailed mathematical derivation from a trigonometric identity. It starts with given expressions for \(\sin\theta\) and \(\cos\theta\) and uses the Pythagorean identity to derive an expression for \(\sin\theta\) in terms of \(x\). The document does not contain a specific question or a question and answer format.
  
  Label: no-question