text: # Thread: Solve this trig equation
  
  1. ## Solve this trig equation
  
  sin(3x) = cos(x + pi/4)
  
  Thanks
  
  2. my attempt
  sin 3x = cos(x+pi/4)
  sin 3x= sin{ pi/2- (x+pi/2)}
  sin 3x= sin( pi/4 - x )
  gives
  3x=pi/4 - x
  
  x=pi/16
  
  3. $\displaystyle \sin(3x) = \cos( x + \frac{\pi}{4} )$
  
  $\displaystyle \sin(3x) = \sin( \frac{\pi}{2} - x - \frac{\pi}{4} )$
  
  $\displaystyle \sin(3x) = \sin( \frac{\pi}{4} - x )$
  
  $\displaystyle 3x = 2n\pi + \frac{\pi}{4} - x$
  
  $\displaystyle x = \frac{(8n+1)\pi}{16}$
  
  Or
  
  $\displaystyle 3x = (2n+1)\pi - \frac{\pi}{4} + x$
  
  $\displaystyle x = \frac{( 8n + 3 )\pi}{8}$
  
  4. where did 2 n come from?
  
  5. Originally Posted by usagi_killer
  where did 2 n come from?
  its because there is no restriction to the domain of x .
  
  6. i have plotted this function, sin(3x) - cos(x + pi/4) = 0, for x = 0 to 2pi
  
  it does have 6 roots for x = 0 to 2 pi
  
  using the sum to product formula, we have this
  
  -2 sin (pi/8 -2x) cos (x + pi/8) = 0
  
  this one form does not generate the 6 roots as shown in the plot, i wonder
explanation: The document is a thread from a forum discussing the solution to a trigonometric equation. It starts with a user posting the equation \( \sin(3x) = \cos(x + \pi/4) \) and asking for help. Another user provides an attempt at solving the equation, and a third user offers a more detailed solution, including the general form of the solution involving the integer \( n \). The thread also includes a discussion about the number of roots within a specific interval and a plot of the function.
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: # Thread: Solve this trig equation
  
  1. ## Solve this trig equation
  
  sin(3x) = cos(x + pi/4)
  
  Thanks
  
  2. my attempt
  sin 3x = cos(x+pi/4)
  sin 3x= sin{ pi/2- (x+pi/2)}
  sin 3x= sin( pi/4 - x )
  gives
  3x=pi/4 - x
  
  x=pi/16
  
  3. $\displaystyle \sin(3x) = \cos( x + \frac{\pi}{4} )$
  
  $\displaystyle \sin(3x) = \sin( \frac{\pi}{2} - x - \frac{\pi}{4} )$
  
  $\displaystyle \sin(3x) = \sin( \frac{\pi}{4} - x )$
  
  $\displaystyle 3x = 2n\pi + \frac{\pi}{4} - x$
  
  $\displaystyle x = \frac{(8n+1)\pi}{16}$
  
  Or
  
  $\displaystyle 3x = (2n+1)\pi - \frac{\pi}{4} + x$
  
  $\displaystyle x = \frac{( 8n + 3 )\pi}{8}$
  
  4. where did 2 n come from?
  
  5. Originally Posted by usagi_killer
  where did 2 n come from?
  its because there is no restriction to the domain of x .
  
  6. i have plotted this function, sin(3x) - cos(x + pi/4) = 0, for x = 0 to 2pi
  
  it does have 6 roots for x = 0 to 2 pi
  
  using the sum to product formula, we have this
  
  -2 sin (pi/8 -2x) cos (x + pi/8) = 0
  
  this one form does not generate the 6 roots as shown in the plot, i wonder
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a thread from a forum discussing the solution to a trigonometric equation. It starts with a user posting the equation \( \sin(3x) = \cos(x + \pi/4) \) and asking for help. Another user provides an attempt at solving the equation, and a third user offers a more detailed solution, including the general form of the solution involving the integer \( n \). The thread also includes a discussion about the number of roots within a specific interval and a plot of the function.
  
  Label: question-and-answer