Bonjour,
Formules à utiliser:
[tex](a\pm b)^2=a^2 \pm ab+b^2}\\\\sin\ (a \pm\ b)=sin\ (a)*cos(b)\ \pm\ cos\ (a)*sin\ (b)\\\\cos\ (\dfrac{2\pi}{3})=\dfrac{-1}{2} \\\\sin \ (\dfrac{2\pi}{3})=\dfrac{\sqrt{3}}{2}\\[/tex]
[tex]sin \ (x+\dfrac{2\pi}{3})=-\dfrac{sin\ (x)}{2} +\dfrac{\sqrt{3} cos\ (x)}{2} \\\\sin \ ^2\ (x+\dfrac{2\pi}{3})=(-\dfrac{sin\ (x)}{2} +\dfrac{\sqrt{3} cos\ (x)}{2} )^2\\\\=\dfrac{sin\ ^2 (x)}{4}-2*\dfrac{1}{2} *\dfrac{\sqrt{3}}{2}*sin(x)*cos(x) +(\dfrac{\sqrt{3}}{2}*cos(x) ) ^2\\\\-----------------\\sin \ (x-\dfrac{2\pi}{3})=-\dfrac{sin\ (x)}{2} -\dfrac{\sqrt{3} cos\ (x)}{2} \\\\sin \ ^2\ (x-\dfrac{2\pi}{3})=(-\dfrac{sin\ (x)}{2} -\dfrac{\sqrt{3} cos\ (x)}{2} )^2\\[/tex]
[tex]=\dfrac{sin\ ^2 (x)}{4}+2*\dfrac{1}{2} *\dfrac{\sqrt{3}}{2}*sin(x)*cos(x) +(\dfrac{\sqrt{3}}{2}*cos(x) ) ^2\\[/tex]
[tex]sin \ ^2\ (x)+sin \ ^2\ (x+\dfrac{2\pi}{3})+sin \ ^2\ (x-\dfrac{2\pi}{3})\\\\=sin^2(x)+\dfrac{sin^2(x)}{2}+\dfrac{3}{2}cos^2(x)\\\\=\dfrac{3}{2}*(sin^2(x)+cos^2(x))\\\\=\dfrac{3}{2}[/tex]
