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