Find $\cos (2 A)$, if $\cos (A)=-\frac{21}{29}$, and $A$ is in quadrant 3
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So, the final answer is $\boxed{0.04875148632580273}$.

Steps

Step 1 ：We are given that $\cos(A) = -\frac{21}{29}$ and that $A$ is in quadrant 3.

Step 2 ：We are asked to find $\cos(2A)$.

Step 3 ：We know that the formula for $\cos(2A)$ is $2\cos^2(A) - 1$.

Step 4 ：Substituting $\cos(A) = -\frac{21}{29}$ into the formula, we get $\cos(2A) = 2(-\frac{21}{29})^2 - 1$.

Step 5 ：Solving this, we find that $\cos(2A) = 0.04875148632580273$.

Step 6 ：Since $A$ is in quadrant 3, $\cos(2A)$ will also be negative.

Step 7 ：So, the final answer is $\boxed{0.04875148632580273}$.