Уравнение вида a sin x + b cos x = c. Задача 15

ОТВЕТ: \(\arcsin \dfrac{2}{{\sqrt {34} }}-\arcsin \dfrac{5}{{\sqrt {34} }} + 2\pi k;\,\,\,\pi -\arcsin \dfrac{2}{{\sqrt {34} }}-\arcsin \dfrac{5}{{\sqrt {34} }} + 2\pi k;\,\,\,k \in Z.\)

\(3\sin x + 5\cos x = 2.\)

Уравнение вида:  \(a\sin x + b\cos x = c,\;\;\)где \(\;a = 3,\,\,\,\,b = 5,\;\;c = 2.\)

Разделим обе части исходного уравнения на  \(\sqrt {{a^2} + {b^2}} ,\;\) то есть на  \(\sqrt {{3^2} + {5^2}}  = \sqrt {34} .\)

\(\dfrac{3}{{\sqrt {34} }}\sin x + \dfrac{5}{{\sqrt {34} }}\cos x = \dfrac{2}{{\sqrt {34} }}.\)

Так как  \({\left( {\dfrac{3}{{\sqrt {34} }}} \right)^2} + {\left( {\dfrac{5}{{\sqrt {34} }}} \right)^2} = 1,\)  то пусть  \(\cos \varphi  = \dfrac{3}{{\sqrt {34} }},\,\,\,\sin \varphi  = \dfrac{5}{{\sqrt {34} }}.\)  Тогда уравнение примет вид:

\(\dfrac{3}{{\sqrt {34} }}\sin x + \dfrac{5}{{\sqrt {34} }}\cos x = \dfrac{2}{{\sqrt {34} }}\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\cos \varphi \sin x + \sin \varphi \cos x = \dfrac{2}{{\sqrt {34} }}\,\,\,\,\,\,\,\, \Leftrightarrow \)

\( \Leftrightarrow \,\,\,\,\,\,\,\sin \left( {x + \varphi } \right) = \dfrac{2}{{\sqrt {34} }}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x + \varphi  = \arcsin \dfrac{2}{{\sqrt {34} }} + 2\pi k,\,\,\,\,\,\,}\\{x + \varphi  = \pi -\arcsin \dfrac{2}{{\sqrt {34} }} + 2\pi k}\end{array}} \right.\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = \arcsin \dfrac{2}{{\sqrt {34} }}-\varphi  + 2\pi k,\,\,\,\,\,\,\,\,\,}\\{x = \pi -\arcsin \dfrac{2}{{\sqrt {34} }}-\varphi  + 2\pi k,}\end{array}} \right.\,\,\,\,k \in Z.\)

Так как  \(\sin \varphi  = \dfrac{5}{{\sqrt {34} }},\)  то   \(\varphi  = \arcsin \dfrac{5}{{\sqrt {34} }}\)  и  

\(\left[ {\begin{array}{*{20}{c}}{x = \arcsin \dfrac{2}{{\sqrt {34} }}-\arcsin \dfrac{5}{{\sqrt {34} }} + 2\pi k,\,\,\,\,\,\,\,\,\,}\\{x = \pi -\arcsin \dfrac{2}{{\sqrt {34} }}-\arcsin \dfrac{5}{{\sqrt {34} }} + 2\pi k,}\end{array}} \right.\,\,\,\,\,k \in Z.\)

Ответ:  \(\arcsin \dfrac{2}{{\sqrt {34} }}-\arcsin \dfrac{5}{{\sqrt {34} }} + 2\pi k;\,\,\,\pi -\arcsin \dfrac{2}{{\sqrt {34} }}-\arcsin \dfrac{5}{{\sqrt {34} }} + 2\pi k;\,\,\,k \in Z.\)