Question

12cosx-5sinx=-6,5 help me please

Answer 1

$12cosx-5sinx=-6,5; |:13; ; (12^2+5^2=169=13^2)\\ frac{12}{13}cdot cosx- frac{5}{13}cdot sinx=- frac{1}{2} \\ frac{12}{13}=cos beta ; ,; ; frac{5}{13}=sin beta ; ; ; (sin^2 beta +cos^2beta =1)\\tg beta =frac{5}{12} ; ; to ; ; beta =arctgfrac{5}{12}\\cos beta cdot cosx-sin beta cdot sinx=-frac{1}{2}\\cos(x+ beta )=-frac{1}{2}\\x+ beta =pm (pi -frac{pi}{3})+2pi n=pm frac{2pi }{3}+2pi n,; nin Z\\x=pm frac{2pi}{3}-beta +2pi n,; nin Z$

$x=pm frac{2pi}{3}-arctgfrac{5}{12}+2pi n,; nin Z$