Применение основных тригонометрических формул для решения уравнений. Задача 25

ОТВЕТ: \(\pi k;\quad k \in Z.\)

\(\cos 2x + 3\sin x = 1\)

Воспользуемся формулой косинуса двойного угла:

\(\cos 2\alpha  = 1-2{\sin ^2}\alpha .\)

\(\cos 2x + 3\sin x = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,1-2{\sin ^2}x + 3\sin x = 1\,\,\,\,\,\,\, \Leftrightarrow \)

\( \Leftrightarrow \,\,\,\,\,\,\,2{\sin ^2}x-3\sin x = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\sin x\left( {2\sin x-3} \right) = 0\,\,\,\,\,\,\, \Leftrightarrow \)

\( \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\sin x = 0,\,\,\,\,\,\,\,\,\,\,}\\{2\sin x-3 = 0}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = \pi k,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\sin x = \dfrac{3}{2}\,\, \notin \,\left[ {-1;1} \right]}\end{array}} \right.\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = \pi k,\,\,\,\,\,\,\,\,k\, \in \,Z.\)

Ответ: \(\pi k;\quad k \in Z.\)