13.49=10sinX+25sin(2X-100) resoudre l'equation

Bonjour Toungbor

Résoudre en grades l'équation 13,49 = 10 sinx + 25 sin(2x-100)

[tex]13,49 = 10 \sin x + 25 \sin(2x-100)\\13,49 = 10 \sin x - 25 \sin(100-2x)\\13,49 = 10 \sin x - 25 \cos(2x)\\13,49 = 10 \sin x - 25 (1-2\sin^2x)\\13,49 = 10 \sin x - 25+50\sin^2x\\50\sin^2x+10 \sin x- 25-13,49=0\\50\sin^2x+10 \sin x - 38,49=0\\\\Soit\ X=\sin x\\\\Alors\ 50X^2+10X-38,49=0[/tex]

[tex]\Delta=10^2-4\times50\times(-38,49)=100+7698=7798\ \textgreater \ 0\\\\X_1=\dfrac{-10-\sqrt{7798}}{100}\approx-0,983\\\\X_2=\dfrac{-10+\sqrt{7798}}{100}\approx0,783[/tex]

D'où,

[tex]\sin x=-0,983\\\\\Longrightarrow \boxed{x=-88,2\ gr+400k\ gr\ \ ou\ \ x=288,8\ gr+400k\ gr\ \ (k\in\mathbb{Z})}\\\\ou\\\\\sin x=0,783\\\\\Longrightarrow \boxed{x=57,26\ gr+400k\ gr\ \ ou\ \ x=142,74\ gr+400k\ gr\ \ (k\in\mathbb{Z})}[/tex]