Solving for x,
x = (1/7)*(4+2sinx). Now, note that
|x-y| = (2/7)*|sinx-siny| = (2/7)*|cos(z)||x-y| <= (2/7)*|x-y|, where x < z < y. (you will need to fill in the details)
Therefore, we see that the function T(x) = (1/7)*(4+2sinx) is a contraction mapping on the real numbers.
By the contraction mapping theorem, the equation has a unique real solution x.