Let's solve the sine equation.
1. Express sine function in the left side of equation:
[tex] 2\sin (4x)+6=5,\\ 2\sin (4x)=5-6,\\ 2\sin (4x)=-1,\\ \\ \sin (4x)=-\dfrac{1}{2} [/tex].
2. Use the genereal solution to get the solution of your equation:
[tex] 4x=(-1)^n\arcsin \left(-\dfrac{1}{2}\right) +2\pi n [/tex], where [tex] n\in Z [/tex].
3. Find [tex] \arcsin \left(-\dfrac{1}{2}\right) [/tex]:
[tex] \arcsin \left(-\dfrac{1}{2}\right)=-\dfrac{\pi}{3} [/tex].
4. Substitute part 3 into part 2 and express x:
[tex] 4x=(-1)^n \left(-\dfrac{\pi}{3}\right) +2\pi n [/tex], where [tex] n\in Z, [/tex]
[tex]x=(-1)^{n+1}\cdot \dfrac{\pi}{12}+\dfrac{\pi n}{2} [/tex], where [tex] n\in Z[/tex].
5. Solutions of your equation are:
[tex]x=(-1)^{n+1}\cdot \dfrac{\pi}{12}+\dfrac{\pi n}{2} [/tex], where [tex] n\in Z [/tex].
