Question

Решить уравнение:
tg8x = tg11x

Answer 1

$tg(alpha)-tg(beta)=frac{sin(alpha-beta)}{cos(alpha)*cos(beta)}$

$tg(8x)=tg(11x)\\ tg(11x)-tg(8x)=0\\ frac{sin(11x-8x)}{cos(11x)*cos(8x)}=0\\ frac{sin(3x)}{cos(11x)*cos(8x)}=0\\ left { {{sin(3x)=0} atop {cos(11x)neq0 and cos(8x)neq0} right. \\ left { {{3x=pi n, nin Z} atop {11xneqfrac{pi}{2}+pi k, kin Z and 8xneqfrac{pi}{2}+pi p, pin Z} right. \\ left { {{x=frac{pi n}{3}, nin Z} atop {xneqfrac{pi}{22}+frac{pi k}{11}, kin Z and xneqfrac{pi}{16}+frac{pi p}{8}, pin Z} right.$

$x=frac{pi n}{3}, nin Z$

Ответ: $frac{pi n}{3}, nin Z$