Find $\cos (2 A)$, if $\cos (A)=-\frac{21}{29}$, and $A$ is in quadrant 3

Step 1 ：Given that \(\cos(A) = -\frac{21}{29}\) and \(A\) is in quadrant 3

Step 2 ：We know that the cosine of an angle in the third quadrant is negative

Step 3 ：We can use the formula for \(\cos(2A)\), which is \(2\cos^2(A) - 1\)

Step 4 ：Substitute the given value of \(\cos(A)\) into this formula to find \(\cos(2A)\)

Step 5 ：\(\cos(2A) = 2(-\frac{21}{29})^2 - 1\)

Step 6 ：Simplify to get \(\cos(2A) = 0.04875148632580273\)

Step 7 ：Final Answer: \(\boxed{0.04875148632580273}\)