Question

Solve for x in the following:​

Answer 1

`\boxed{\boxed{ \red{\: x = \begin{cases} (\pi)/(8) + 7 + (k\pi)/(2) \\ - 6 + (\pi)/(12) + (k\pi)/(3) \end{cases}}}}`

Explanation:

to understand this

you need to know about:

• trigonometry equation
• PEMDAS

tips and formulas:

• 
  `\cos(t) = \sin( (\pi)/(2) - t )`
• 
  `\sin(t) - \sin(s) = 2 \cos( (t + s)/(2) ) \sin( (t - s)/(2) )`

let's solve:

1. 
   `\sf use \: first \: formula : \\ \sin(5x + 4) = \sin( (\pi)/(2) - (5 x + 4) )\\ \sin(5x + 4) = \sin( (\pi)/(2) - 5 x - 4) \\`
2. 
   `\sf move \: the \: expresson \: to \: left \: side \: and \: change \: the \: sign : \\ \sin(5x + 4) - \sin( (\pi)/(2) - 5 x - 4) = 0\\`
3. 
   `\sf use \: 2nd \: formula : \\ 2 \cos( (8x - 56 + \pi)/(4) ) \sin( (12x + 72 - \pi)/(4) ) = 0`
4. 
   `\sf divide \: both \: sides \: by \: 2 : \\ \cos( (8x - 56 + \pi)/(4) ) \sin( (12x + 72 - \pi)/(4) ) = 0`
5. 
   `\sf separate \: the \: equation: \\ \cos( (8x - 56 + \pi)/(4) ) = 0 \\ \sin( (12x + 72 - \pi)/(4) ) = 0`

therefore

`\therefore \: x = \begin{cases} (\pi)/(8) + 7 + (k\pi)/(2) \\ - 6 + (\pi)/(12) + (k\pi)/(3) \end{cases}`