Question

Решите пожалуйста!!!

Answer 1

$4Sin^{3}x=Cos(x-\frac{5\pi }{2})\\\\4Sin^{3}x=Cos(2\pi+\frac{\pi }{2}-x)\\\\4Sin^{3}x=Cos(\frac{\pi }{2}-x)\\\\4Sin^{3}x=Sinx\\\\4Sin^{3}x-Sinx=0\\\\Sinx(4Sin^{2}x-1)=0\\\\1)Sinx=0\\\\x=\pi n,n\in Z$

$2)4Sin^{2}x-1=0\\\\Sin^{2}x=\frac{1}{4}\\\\Sinx=\pm \frac{1}{2}\\\\Sinx=\frac{1}{2}\\\\\left[\begin{array}{ccc}x=arcSin\frac{1}{2}+2\pi n,n\in Z\\x=\pi-arcSin\frac{1}{2}+2\pi n,n\in Z \end{array}\right\\\\\left[\begin{array}{ccc}x=\frac{\pi }{6} +2\pi n,n\in Z \\x=\frac{5\pi }{6}+2\pi n,n\in Z \end{array}\right\\\\n=1\Rightarrow x=\frac{\pi }{6}+2\pi=\frac{13\pi }{6}\\\\Sinx=-\frac{1}{2}$

$\left[\begin{array}{ccc}x=-\frac{\pi }{6}+2\pi n,n\in Z \\x=\frac{7\pi }{6}+2\pi n,n\in Z \end{array}\right\\\\n=1\Rightarrow x=-\frac{\pi }{6}+2\pi=\frac{11\pi }{6}\\\\Otvet:\boxed{2\pi ;\frac{11\pi }{6};\frac{13\pi }{6}}$