shukronaikromova46
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Solve for x in the following:​

Solve For X In The Following class=

Sagot :

Answer:

[tex] \boxed{\boxed{ \red{\: x = \begin{cases} \frac{\pi}{8} + 7 + \frac{k\pi}{2} \\ - 6 + \frac{\pi}{12} + \frac{k\pi}{3} \end{cases}}}}[/tex]

Step-by-step explanation:

to understand this

you need to know about:

  • trigonometry equation
  • PEMDAS

tips and formulas:

  • [tex] \cos(t) = \sin( \frac{\pi}{2} - t ) [/tex]
  • [tex] \sin(t) - \sin(s) = 2 \cos( \frac{t + s}{2} ) \sin( \frac{t - s}{2} ) [/tex]

let's solve:

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

therefore

[tex] \therefore \: x = \begin{cases} \frac{\pi}{8} + 7 + \frac{k\pi}{2} \\ - 6 + \frac{\pi}{12} + \frac{k\pi}{3} \end{cases}[/tex]

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