Question

Sin x + sin 5x = sin 3x с подробным решением​

Answer 1

$Sinx+Sin5x=Sin3x\\\\2Sin\frac{x+5x}{2}Cos\frac{x-5x}{2}=Sin3x\\\\2Sin3xCos2x-Sin3x=0\\\\Sin3x(2Cos2x-1)=0\\\\\left[\begin{array}{ccc}Sin3x=0\\2Cos2x-1=0\end{array}\right\\\\\left[\begin{array}{ccc}3x=\pi n,n\in Z \\Cos2x=\frac{1}{2} \end{array}\right\\\\\left[\begin{array}{ccc}x=\frac{\pi n }{3},n\in Z \\2x=\pm arcCos\frac{1}{2}+2\pi n,n\in Z \end{array}\right\\\\\left[\begin{array}{ccc}x=\frac{\pi n }{3},n\in Z \\2x=\pm \frac{\pi }{3}+2\pi n,n\in Z \end{array}\right$

$\left[\begin{array}{ccc}x=\frac{\pi n }{3},n\in Z \\x=\pm \frac{\pi }{6}+\pi n,n\in Z \end{array}\right$