Use identities to find values of the sine and cosine functions of the function for the angle measure. $2 \theta$, given $\sin \theta=-\frac{\sqrt{6}}{7}$ and $\cos \theta>0$ \[ \cos 2 \theta= \] (Use fractions or pi for any numbers in the expression.)

Step 1 ：We are given that \(\sin \theta = -\frac{\sqrt{6}}{7}\) and \(\cos \theta > 0\).

Step 2 ：We know that the formula for cosine of double angle is given by: \(\cos 2 \theta = 1 - 2 \sin^2 \theta\).

Step 3 ：Substitute \(\sin \theta = -\frac{\sqrt{6}}{7}\) into the formula to find \(\cos 2 \theta\).

Step 4 ：After calculation, we find that the value of \(\cos 2 \theta\) is approximately 0.755.

Step 5 ：So, the final answer is \(\boxed{0.755}\).