text: # Trigonometric Equations: Problems with Solutions
  
  By Denitsa Dimitrova (Bulgaria)
  Problem 1
  $\cos(2x)-7\sin x-4=0$
  Problem 2
  $\cos2x-5\cos x+3=0,\ \ x\in \left[0,\frac{\pi}{2}\right]$
  Problem 3
  $\cos 2x-5\sin x-3=0$
  Problem 4
  $\cos 2x+6\sin x-5=0, x\in(0, \pi]$
  Problem 5
  $5\sin x-2\cos^2x-1=0$
  
  $x\in\left[0,\frac{\pi}{2}\right]$
  Problem 6
  $3+\cos 2x+3\sqrt{2}\cos x=0$
  
  $x\in(0,\pi)$
  Problem 7
  $\cos2x+\frac{1}{\sqrt{1+\cot^2(x)}}=0$
  Problem 8
  $\sin^4x+\cos^4x=\frac{5}{8}, x\in [0, \pi]$
  Problem 9
  $\sin^3x-3\sin^2 x+3\sin x=1$
  $x\in [0, \pi]$
  Problem 10
  $(\cos2x-1)\cot^2x=-3\sin x$
  $x\in[0,\pi)$
  
  Problem 11
  $3\cos16x+8\sin^22x\cdot \cos^22x-5=0$
  $x\in[0,\pi]$
  Problem 12
  $\sin^2x-3\sin x\cos x+2\cos^2x=0$
  $x\in [0,\frac{\pi}{2}]$
  Problem 13
  $\sin^2x-(\sqrt{3}+1)\sin x \cos x +\sqrt{3}\cos^2x=0$
  $x \in [0, \pi]$
  Problem 14
  $\cos x+ \sin x = \frac{\sqrt{2}}{2}$
  $x\in[0,\pi]$
  Problem 15
  $\sin 2x-\sqrt{3}\cos 2x=-\sqrt{3}$
  Problem 16
  $\sin 2x-\cos x=0$
  Problem 17
  $\sin 3x =\sin x$
  $x\in[0,2\pi]$
  Problem 18
  $\sin x+\sin 3x=\sin 2x+\sin 4x$
  $x\in[0,\pi]$
  Problem 19
  $\cos x + \cos 2x+\cos 3x = 0$
  Problem 20
  $\sin 4x+\sin20x-\sin12x=0$
  Problem 21
  $\sin 3x+\cos(x-\frac{\pi}{6})=2$
  
  Correct:
  Wrong:
  Unsolved problems:
  Feedback   Contact email:
explanation: The document is a collection of trigonometric equations presented as problems, each with a specific equation and a defined domain for the variable \( x \). The document does not provide solutions or answers to these problems. It ends with placeholders for correct, wrong, and unsolved problems, indicating that this is likely a worksheet or practice set for students.
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: # Trigonometric Equations: Problems with Solutions
  
  By Denitsa Dimitrova (Bulgaria)
  Problem 1
  $\cos(2x)-7\sin x-4=0$
  Problem 2
  $\cos2x-5\cos x+3=0,\ \ x\in \left[0,\frac{\pi}{2}\right]$
  Problem 3
  $\cos 2x-5\sin x-3=0$
  Problem 4
  $\cos 2x+6\sin x-5=0, x\in(0, \pi]$
  Problem 5
  $5\sin x-2\cos^2x-1=0$
  
  $x\in\left[0,\frac{\pi}{2}\right]$
  Problem 6
  $3+\cos 2x+3\sqrt{2}\cos x=0$
  
  $x\in(0,\pi)$
  Problem 7
  $\cos2x+\frac{1}{\sqrt{1+\cot^2(x)}}=0$
  Problem 8
  $\sin^4x+\cos^4x=\frac{5}{8}, x\in [0, \pi]$
  Problem 9
  $\sin^3x-3\sin^2 x+3\sin x=1$
  $x\in [0, \pi]$
  Problem 10
  $(\cos2x-1)\cot^2x=-3\sin x$
  $x\in[0,\pi)$
  
  Problem 11
  $3\cos16x+8\sin^22x\cdot \cos^22x-5=0$
  $x\in[0,\pi]$
  Problem 12
  $\sin^2x-3\sin x\cos x+2\cos^2x=0$
  $x\in [0,\frac{\pi}{2}]$
  Problem 13
  $\sin^2x-(\sqrt{3}+1)\sin x \cos x +\sqrt{3}\cos^2x=0$
  $x \in [0, \pi]$
  Problem 14
  $\cos x+ \sin x = \frac{\sqrt{2}}{2}$
  $x\in[0,\pi]$
  Problem 15
  $\sin 2x-\sqrt{3}\cos 2x=-\sqrt{3}$
  Problem 16
  $\sin 2x-\cos x=0$
  Problem 17
  $\sin 3x =\sin x$
  $x\in[0,2\pi]$
  Problem 18
  $\sin x+\sin 3x=\sin 2x+\sin 4x$
  $x\in[0,\pi]$
  Problem 19
  $\cos x + \cos 2x+\cos 3x = 0$
  Problem 20
  $\sin 4x+\sin20x-\sin12x=0$
  Problem 21
  $\sin 3x+\cos(x-\frac{\pi}{6})=2$
  
  Correct:
  Wrong:
  Unsolved problems:
  Feedback   Contact email:
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a collection of trigonometric equations presented as problems, each with a specific equation and a defined domain for the variable \( x \). The document does not provide solutions or answers to these problems. It ends with placeholders for correct, wrong, and unsolved problems, indicating that this is likely a worksheet or practice set for students.
  
  Label: multiple-question