Question

How can i solve sin(7x)=sin(5x)?

Answer 1

   
$\displaystyle \\ \texttt{The equation has three solutions.} \\ \\ S1: \\ \sin(7x) = \sin(5x) \\ 7x = 5x \\ 7x-5x=0 \\ 2x=0 \\ \boxed{x_1 = 0^o} \\ Check: \\ \sin(7\cdot 0)= \sin(0) = 0~~and~~\sin(5\cdot 0)= \sin(0) = 0$


$\displaystyle S2: \\ \sin(7x) = \sin(5x) \\ \sin(180-\alpha)=\sin(\alpha) ~~\Longrightarrow~~ \sin(90+\beta)=\sin(90-\beta) \\ \Longrightarrow~~ \frac{7x + 5x}{2} = 90^o \\ \\ \frac{12x}{2} = 90^o \\ \\ 6x = 90^o \\ \\ x_2 = \frac{90}{6} \\ \\ \boxed{x_2 = 15^o }\\ \\ Check: \\ \sin(7\cdot 15^o) = \sin(105^o)=\sin(180^o-105^o)=\sin(75^o)=\sin(5\cdot 15^o)$


$\displaystyle S3: \\ \sin(7x) = \sin(5x) \\ \sin(180+\alpha)=\sin(360-\alpha) ~~\Longrightarrow~~ \sin(270+\beta)=\sin(270-\beta) \\ \Longrightarrow~~ \frac{7x + 5x}{2} = 270^o \\ \\ \frac{12x}{2} = 270^o \\ \\ 6x = 270^o \\ \\ x_3 = \frac{270}{6} \\ \\ \boxed{x_3 = 45^o }\\ \\ Check: \\ \sin(7\cdot 45^o) = \sin(315^o)=\sin(360^o-45^o)=\sin(180+45^o)= \\ = \sin(225^o) = \sin(5 \cdot 45^0)$