Question

решите уравнеие ctgx-4=5tng​

Answer 1

$displaystyle ctgx-4=5tgx\\text{OD3} :quad left[begin{array}{ccc}sinxneq0\cosxneq0end{array}rightquad rightarrowquad left[begin{array}{ccc}xneqpi n;,,nin Z\xneqfrac{pi}2+pi n;,,nin Zend{array}rightquad rightarrowquad xneqfrac{pi n}2;,,nin Z\\\frac{1}{tgx}-4=5tgx\\1-4tgx=5tg^2x\\5tg^2x+4tgx-1=0\\tgx=t,quad tin R\\5t^2+4t-1=0\\text{D}=16+20=36=6^2\\t_1=frac{-4+6}{10}=frac{2}{10}=frac{1}5\\t_2=frac{-4-6}{10}=-frac{10}{10}=-1$

$displaystyle left[begin{array}{ccc}displaystyle tgx=frac{1}5\\tgx=-1end{array}rightquadrightarrowquadboxed{left[begin{array}{ccc}displaystyle x=arctgbigg(frac{1}5bigg)+pi n;,,nin Z\\displaystyle x=-frac{pi}4+pi n;,,nin Zend{array}right}$