Question

Тригонометрическое уравнение, помогите:
tg3x=(2+sqrt3)tgx

Answer 1

$tg3x=(2+sqrt3)tgx\\ frac{3tgx-tg^3x}{1-3tg^2x} =(2+sqrt3)tgx\\t=tgx; ; to ; ; frac{3t-t^3}{1-3t^2} =(2+sqrt3)t\\3t-t^3=(2+sqrt3)cdot tcdot (1-3t^2)\\3t-t^3=(2+sqrt3)cdot t-3(2+sqrt3)t^3\\(3-2+sqrt3)cdot t+(6+3sqrt3-1)t^3=0\\tcdot ((1+sqrt3)+(5+3sqrt3)t^2)=0\\tcdot (1+sqrt3+(5+3sqrt3)t^2)=0\\Tak; kak; ; 1+sqrt3+(5+3sqrt3)t^2 textgreater 0; ,; to; ; mnozitel; ; t=0; .\\tgx=0; ; to ; ; x=pi n,; nin Z$