Question

Solve the following equations.
$si{n}^4\frac{x}{3}+co{s}^4\frac{x}{3}=\frac{5}{8}.$

Answer 1

$si{n}^4\frac{x}{3}+co{s}^4\frac{x}{3}=\frac{5}{8}.$

$\Rightarrow (si{n}^{2}x/3{)}^2+(co{s}^{2}x/3{)}^2+2si{n}^{2}x/3.co{s}^{2}x/3-2si{n}^{2}x/3.co{s}^{2}x/3=5/8$

$\Rightarrow (si{n}^{2}x/3+co{s}^{2}x/3{)}^2-2si{n}^{2}x/3.co{s}^{2}x/3=5/8$

$\Rightarrow 1-2si{n}^{2}x/3co{s}^{2}x/3=5/8$

$\Rightarrow 2si{n}^{2}x/3co{s}^{2}x/3=3/8$

$\Rightarrow \frac{si{n}^{2}2x/3}{2}=3/8$

$\Rightarrow \frac{1-cos4x/3}{4}=\frac{3}{8}$

$\Rightarrow cos4x/3=-1/2$

$4x/3=\pm 2\pi /3+2n\pi$

$\therefore x=\frac{3x\pi }{2}\pm \frac{\pi }{2}$

$nϵ$ integer