Question

Solve the following equations.
$si{n}^{2}x+si{n}^{2}2x+si{n}^{2}3x=\frac{3}{2}.$

Answer 1

$si{n}^{2}x+si{n}^{2}2x+si{n}^{2}3x=3/2$

multiply the entire equation by 2

$\Rightarrow 2si{n}^{2}x+2si{n}^{2}2x+2si{n}^{2}3x=3$

$\Rightarrow 1-[cos2x+cos4x+cos6x]=3$

$\Rightarrow 3-[cos2x+cos4x+cos6x]=3$

$\Rightarrow cos2x+cos4x+cos6x=0$

$\Rightarrow cos2x+cos6x+cos4x=0$

$\Rightarrow 2cos4x.cos2x+cos4x=0$

$\Rightarrow cos4x(1+2cos2x)=0$

$\therefore cos4x=0$ or $1+2cos2x=0$

$\Rightarrow 4x=(2n+1)\pi /2$ or $cos2x=-1/2$

$\Rightarrow x=(2n+1)\pi /8$ or $x=m\pi \pm \pi /3$

$nϵ$ integer

$mϵ$ integer