Answer
$$x=\frac{5\pi }{4}+2\pi n,\:x=\frac{7\pi }{4}+2\pi nE$$
Work Step by Step
We solve the equation using the properties of trigonometric functions. Note, there is a general solution since trigonometric identities go up and down and this can pass through a given value of y many times. Solving this, we find:
$$\sin \left(x\right)+\sqrt{2}=-\sin \left(x\right) \\ \sin \left(x\right)=-\frac{\sqrt{2}}{2} \\ x=\frac{5\pi }{4}+2\pi n,\:x=\frac{7\pi }{4}+2\pi n$$
